Extrema in 2 Variables
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[{"Name":"Extrema in 2 Variables","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Extrema in 2 Variables","Duration":"17m 23s","ChapterTopicVideoID":8721,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8721.jpeg","UploadDate":"2020-02-26T05:56:11.5830000","DurationForVideoObject":"PT17M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this clip, we\u0027ll learn about extrema and"},{"Start":"00:03.360 ","End":"00:07.905","Text":"also something called saddle points of functions of 2 variables."},{"Start":"00:07.905 ","End":"00:10.515","Text":"All the terms will be explained."},{"Start":"00:10.515 ","End":"00:14.430","Text":"I think it\u0027s best to start right away with an example."},{"Start":"00:14.430 ","End":"00:18.460","Text":"I\u0027m going to take the following example."},{"Start":"00:18.980 ","End":"00:23.790","Text":"We\u0027ll take the function of 2 variables, x and y,"},{"Start":"00:23.790 ","End":"00:32.040","Text":"to be x cubed plus y cubed, minus 3xy."},{"Start":"00:32.040 ","End":"00:35.825","Text":"Now, we find out these extrema and saddle points,"},{"Start":"00:35.825 ","End":"00:39.875","Text":"which I will shortly clarify, in various steps."},{"Start":"00:39.875 ","End":"00:44.160","Text":"There\u0027s a Step 0 or preliminary step."},{"Start":"00:47.060 ","End":"00:50.600","Text":"What we do in this preliminary step is to compute"},{"Start":"00:50.600 ","End":"00:53.660","Text":"all the partial derivatives up to second-order."},{"Start":"00:53.660 ","End":"00:55.745","Text":"There\u0027s 2 first-order derivatives,"},{"Start":"00:55.745 ","End":"01:00.665","Text":"fx of x and y. I won\u0027t bother writing all the brackets x, y each time."},{"Start":"01:00.665 ","End":"01:03.380","Text":"There\u0027s also partial derivative of f with respect to y."},{"Start":"01:03.380 ","End":"01:08.899","Text":"For second-order, there\u0027s f with respect to x twice,"},{"Start":"01:08.899 ","End":"01:11.345","Text":"there\u0027s f with respect to x then y."},{"Start":"01:11.345 ","End":"01:13.820","Text":"There\u0027s a theorem that this mostly,"},{"Start":"01:13.820 ","End":"01:17.750","Text":"something very unusual circumstances is the same as yx,"},{"Start":"01:17.750 ","End":"01:19.685","Text":"so just leave it like this."},{"Start":"01:19.685 ","End":"01:23.075","Text":"Then there\u0027s f with respect to y and with respect to y."},{"Start":"01:23.075 ","End":"01:28.210","Text":"Let\u0027s do this and then you\u0027ll see why in the following steps."},{"Start":"01:28.210 ","End":"01:31.305","Text":"This is just all technical work."},{"Start":"01:31.305 ","End":"01:35.870","Text":"F with respect to x is the partial derivative with respect to x,"},{"Start":"01:35.870 ","End":"01:37.565","Text":"which means y is a constant."},{"Start":"01:37.565 ","End":"01:41.225","Text":"This is 3x squared, this is nothing,"},{"Start":"01:41.225 ","End":"01:46.000","Text":"and this would be minus 3y."},{"Start":"01:46.000 ","End":"01:55.980","Text":"F with respect to y is equal to similarly 3y squared minus 3x."},{"Start":"01:55.980 ","End":"01:57.885","Text":"Here, x is a constant."},{"Start":"01:57.885 ","End":"02:02.640","Text":"Now, we get the second-order derivative, so let\u0027s see."},{"Start":"02:02.640 ","End":"02:07.055","Text":"Fxx will equal the derivative of this with respect to x."},{"Start":"02:07.055 ","End":"02:11.390","Text":"That will just give me 6x."},{"Start":"02:11.390 ","End":"02:16.485","Text":"Then we have the mixed fxy,"},{"Start":"02:16.485 ","End":"02:19.190","Text":"and you should get the same thing if you do this with"},{"Start":"02:19.190 ","End":"02:22.040","Text":"respect to x or this with respect to y."},{"Start":"02:22.040 ","End":"02:24.050","Text":"In any event in each case,"},{"Start":"02:24.050 ","End":"02:27.620","Text":"we get just minus 3."},{"Start":"02:27.620 ","End":"02:30.725","Text":"There is another derivative fyy,"},{"Start":"02:30.725 ","End":"02:33.395","Text":"which is this with respect to y,"},{"Start":"02:33.395 ","End":"02:37.250","Text":"and that comes out to be 6y."},{"Start":"02:37.250 ","End":"02:39.920","Text":"For consistency, I\u0027ll write the x,"},{"Start":"02:39.920 ","End":"02:41.620","Text":"y in each of them."},{"Start":"02:41.620 ","End":"02:46.260","Text":"Next, we\u0027ll have Step 1,"},{"Start":"02:46.260 ","End":"02:48.735","Text":"which is I call it,"},{"Start":"02:48.735 ","End":"02:58.820","Text":"finding the critical points."},{"Start":"02:58.820 ","End":"03:02.510","Text":"The question is, what is the critical point in functions of 2 variables?"},{"Start":"03:02.510 ","End":"03:06.225","Text":"Well, it\u0027s similar to critical points in 1 variable."},{"Start":"03:06.225 ","End":"03:09.740","Text":"There, we just said that where the derivative is 0,"},{"Start":"03:09.740 ","End":"03:11.380","Text":"that\u0027s a critical point."},{"Start":"03:11.380 ","End":"03:13.670","Text":"Well, here we have 2 derivatives,"},{"Start":"03:13.670 ","End":"03:15.800","Text":"first-order derivatives, this and this,"},{"Start":"03:15.800 ","End":"03:17.990","Text":"and we want both of them to be 0."},{"Start":"03:17.990 ","End":"03:19.850","Text":"I have 2 equations,"},{"Start":"03:19.850 ","End":"03:22.505","Text":"f with respect to x equals 0,"},{"Start":"03:22.505 ","End":"03:25.900","Text":"and f with respect to y equals 0."},{"Start":"03:25.900 ","End":"03:30.260","Text":"In general, I won\u0027t be bothering with all this x, y."},{"Start":"03:30.260 ","End":"03:32.000","Text":"It\u0027s understood."},{"Start":"03:32.000 ","End":"03:34.850","Text":"Just an extra writing."},{"Start":"03:34.850 ","End":"03:37.370","Text":"Now, this is in general, in our case,"},{"Start":"03:37.370 ","End":"03:39.559","Text":"we have the partial derivatives,"},{"Start":"03:39.559 ","End":"03:41.420","Text":"we have 1 here, and 1 here."},{"Start":"03:41.420 ","End":"03:44.150","Text":"That means that both of these have to be 0."},{"Start":"03:44.150 ","End":"03:48.805","Text":"We get 3x squared minus 3y equals 0,"},{"Start":"03:48.805 ","End":"03:52.950","Text":"and 3y squared minus 3x equals 0."},{"Start":"03:52.950 ","End":"03:56.415","Text":"2 equations and 2 unknowns x, y."},{"Start":"03:56.415 ","End":"03:58.755","Text":"We see the 3 is everywhere."},{"Start":"03:58.755 ","End":"04:01.970","Text":"Obviously, we can divide both sides by 3 here and here."},{"Start":"04:01.970 ","End":"04:06.270","Text":"In other words, just remove all the 3\u0027s."},{"Start":"04:06.270 ","End":"04:09.665","Text":"Now what I suggest is take 1 of the equations."},{"Start":"04:09.665 ","End":"04:14.120","Text":"For example, I could take this equation,"},{"Start":"04:14.120 ","End":"04:16.655","Text":"x squared minus y is 0."},{"Start":"04:16.655 ","End":"04:22.699","Text":"If I extract y, that would give me that y equals x squared."},{"Start":"04:22.699 ","End":"04:26.945","Text":"Now that I have y, I can substitute it here."},{"Start":"04:26.945 ","End":"04:30.410","Text":"What I get will be y squared,"},{"Start":"04:30.410 ","End":"04:37.310","Text":"which is x squared squared minus x equals 0."},{"Start":"04:37.310 ","End":"04:45.810","Text":"This gives me x^4 minus x equals 0."},{"Start":"04:45.820 ","End":"04:54.130","Text":"I can take x out, x times x cubed minus 1 equals 0."},{"Start":"04:56.120 ","End":"04:58.920","Text":"This has 2 solutions."},{"Start":"04:58.920 ","End":"05:00.960","Text":"Either x is 0,"},{"Start":"05:00.960 ","End":"05:03.105","Text":"or if x cubed equals 1,"},{"Start":"05:03.105 ","End":"05:05.970","Text":"then x is just the cube root of 1."},{"Start":"05:05.970 ","End":"05:10.505","Text":"There\u0027s only 1 solution unless you talking about complex numbers which were not."},{"Start":"05:10.505 ","End":"05:13.760","Text":"So x equals 0 or x equals 1."},{"Start":"05:13.760 ","End":"05:16.970","Text":"Now, each of these has a corresponding y,"},{"Start":"05:16.970 ","End":"05:21.250","Text":"because we have that y equals x squared."},{"Start":"05:21.250 ","End":"05:24.050","Text":"If x equals 0,"},{"Start":"05:24.050 ","End":"05:28.869","Text":"then y being x squared is 0 squared, which is 0."},{"Start":"05:28.869 ","End":"05:35.460","Text":"If x is 1, then y is 1 squared and it\u0027s equal to 1."},{"Start":"05:35.650 ","End":"05:38.255","Text":"I have 2 critical points,"},{"Start":"05:38.255 ","End":"05:39.560","Text":"this with this and this with this."},{"Start":"05:39.560 ","End":"05:41.165","Text":"Let me just write that down."},{"Start":"05:41.165 ","End":"05:48.650","Text":"We found that the critical points were 0,"},{"Start":"05:48.650 ","End":"05:53.360","Text":"0 and 1, 1."},{"Start":"05:53.360 ","End":"05:59.135","Text":"Now I just realized that I really should have written the question we\u0027re asked to do."},{"Start":"05:59.135 ","End":"06:01.360","Text":"We have to do 2 things."},{"Start":"06:01.360 ","End":"06:03.390","Text":"I want you, A,"},{"Start":"06:03.390 ","End":"06:11.460","Text":"to find the critical points,"},{"Start":"06:11.460 ","End":"06:15.510","Text":"and B, classify them."},{"Start":"06:15.510 ","End":"06:18.920","Text":"Later I\u0027ll write down what this means,"},{"Start":"06:18.920 ","End":"06:22.840","Text":"but I\u0027ll just briefly tell you now there are 3 kinds of critical points;"},{"Start":"06:22.840 ","End":"06:27.005","Text":"maximum, minimum, or saddle points."},{"Start":"06:27.005 ","End":"06:29.360","Text":"We\u0027ve answered Part A of the question,"},{"Start":"06:29.360 ","End":"06:32.180","Text":"in that we\u0027ve found the critical points,"},{"Start":"06:32.180 ","End":"06:35.190","Text":"and now we have to classify them."},{"Start":"06:36.530 ","End":"06:40.320","Text":"Step 2, let me just write something,"},{"Start":"06:40.320 ","End":"06:42.715","Text":"and then I\u0027ll explain it."},{"Start":"06:42.715 ","End":"06:47.980","Text":"We compute something called the discriminant."},{"Start":"06:47.980 ","End":"06:52.480","Text":"The symbol of it,"},{"Start":"06:52.480 ","End":"06:55.390","Text":"sometimes we use the Greek letter Delta,"},{"Start":"06:55.390 ","End":"07:00.080","Text":"sometimes capital Latin letter D. Delta is"},{"Start":"07:00.080 ","End":"07:06.910","Text":"just the Greek for D. This turns out to be a function of x and y."},{"Start":"07:06.910 ","End":"07:15.300","Text":"What we want to do is to evaluate it at the critical points."},{"Start":"07:15.300 ","End":"07:18.400","Text":"What is this discriminant?"},{"Start":"07:18.400 ","End":"07:28.930","Text":"The discriminant is defined to be equal to fxxfyy;"},{"Start":"07:28.930 ","End":"07:37.005","Text":"These are things we\u0027ve computed already, minus fxy squared."},{"Start":"07:37.005 ","End":"07:42.740","Text":"It\u0027s an expression using the second-order partial derivatives."},{"Start":"07:42.740 ","End":"07:49.395","Text":"Of course it\u0027s a function of x and y. Delta of x and y,"},{"Start":"07:49.395 ","End":"07:51.140","Text":"I\u0027ll compute it in a moment."},{"Start":"07:51.140 ","End":"07:53.015","Text":"But I want to tell you what we\u0027re going to do with it,"},{"Start":"07:53.015 ","End":"07:55.055","Text":"now that we\u0027ve found it."},{"Start":"07:55.055 ","End":"08:00.550","Text":"I found a printed version somewhere on the Internet."},{"Start":"08:00.550 ","End":"08:06.735","Text":"Here we are. It\u0027s easier when it\u0027s printed."},{"Start":"08:06.735 ","End":"08:15.720","Text":"Here, we have the definition and it\u0027s pretty much the same as what we had here."},{"Start":"08:16.050 ","End":"08:21.400","Text":"I\u0027ve used Delta, some use D,"},{"Start":"08:21.400 ","End":"08:23.785","Text":"you should get used to seeing both,"},{"Start":"08:23.785 ","End":"08:28.870","Text":"this\u0027s the discriminant and also here it\u0027s written in longhand with the x,"},{"Start":"08:28.870 ","End":"08:32.125","Text":"y and this is shorthand."},{"Start":"08:32.125 ","End":"08:39.295","Text":"These steps are a rule for deciding how to classify the critical point."},{"Start":"08:39.295 ","End":"08:45.865","Text":"We\u0027ll be testing it on 0,0, and on 1,1."},{"Start":"08:45.865 ","End":"08:50.875","Text":"In general, let\u0027s call the critical point a, b."},{"Start":"08:50.875 ","End":"08:56.119","Text":"So a, b could be like in our case 0,0 or 1,1,"},{"Start":"08:56.119 ","End":"08:58.170","Text":"and for each of these points,"},{"Start":"08:58.170 ","End":"09:00.315","Text":"in general, for our critical point,"},{"Start":"09:00.315 ","End":"09:06.370","Text":"we compute the discriminant at that point."},{"Start":"09:06.370 ","End":"09:08.995","Text":"I\u0027ll go over the steps again in a minute,"},{"Start":"09:08.995 ","End":"09:11.530","Text":"but meanwhile, bear with me."},{"Start":"09:11.530 ","End":"09:15.475","Text":"What we do is we have a decision tree,"},{"Start":"09:15.475 ","End":"09:17.274","Text":"we take this value,"},{"Start":"09:17.274 ","End":"09:21.175","Text":"the discriminant at our critical point and say as follows;"},{"Start":"09:21.175 ","End":"09:26.305","Text":"if it\u0027s positive, then there are 2 cases,"},{"Start":"09:26.305 ","End":"09:32.260","Text":"if it\u0027s positive and f_xx is also positive,"},{"Start":"09:32.260 ","End":"09:34.435","Text":"then we have a minimum."},{"Start":"09:34.435 ","End":"09:40.390","Text":"If it\u0027s positive and this f_xx is negative,"},{"Start":"09:40.390 ","End":"09:43.600","Text":"then we have a maximum."},{"Start":"09:43.600 ","End":"09:45.400","Text":"Actually I\u0027ll highlight this."},{"Start":"09:45.400 ","End":"09:49.150","Text":"If it\u0027s positive and this f_xx is positive,"},{"Start":"09:49.150 ","End":"09:54.250","Text":"then we get a minimum and if this is positive,"},{"Start":"09:54.250 ","End":"09:56.155","Text":"but this is negative,"},{"Start":"09:56.155 ","End":"09:58.270","Text":"then we have a maximum."},{"Start":"09:58.270 ","End":"10:01.675","Text":"If the discriminant is negative,"},{"Start":"10:01.675 ","End":"10:03.100","Text":"in regardless of this,"},{"Start":"10:03.100 ","End":"10:05.695","Text":"we have a saddle point."},{"Start":"10:05.695 ","End":"10:09.399","Text":"The fourth case is where it\u0027s equal to 0,"},{"Start":"10:09.399 ","End":"10:11.890","Text":"that\u0027s the don\u0027t know case."},{"Start":"10:11.890 ","End":"10:16.480","Text":"This test is inconclusive and our critical point could be minimum,"},{"Start":"10:16.480 ","End":"10:18.040","Text":"maximum or saddle point."},{"Start":"10:18.040 ","End":"10:19.840","Text":"We either have to say don\u0027t know,"},{"Start":"10:19.840 ","End":"10:23.950","Text":"or use other techniques to determine and classify it."},{"Start":"10:23.950 ","End":"10:29.350","Text":"Let me just reiterate what I said up to now and then soon we\u0027ll do the computations."},{"Start":"10:29.350 ","End":"10:31.900","Text":"The first step is to find the critical points by"},{"Start":"10:31.900 ","End":"10:35.140","Text":"setting the first-order derivatives to 0,"},{"Start":"10:35.140 ","End":"10:38.170","Text":"then we find a certain number of critical points there might not be any,"},{"Start":"10:38.170 ","End":"10:44.450","Text":"there might be several and then we compute the discriminant,"},{"Start":"10:44.450 ","End":"10:47.565","Text":"which is this expression here."},{"Start":"10:47.565 ","End":"10:50.640","Text":"We already did the preliminary phase where we did"},{"Start":"10:50.640 ","End":"10:57.385","Text":"all the computations for the second order derivatives and then we make a decision."},{"Start":"10:57.385 ","End":"11:03.265","Text":"If this discriminant D or Delta is positive,"},{"Start":"11:03.265 ","End":"11:09.925","Text":"then we have to also look at f_xx and also evaluate it at that point,"},{"Start":"11:09.925 ","End":"11:12.595","Text":"if we have this is positive and this is positive minimum,"},{"Start":"11:12.595 ","End":"11:15.850","Text":"this is positive, but this is negative maximum."},{"Start":"11:15.850 ","End":"11:17.710","Text":"If the discriminant comes out negative,"},{"Start":"11:17.710 ","End":"11:19.000","Text":"we don\u0027t need to check anything else,"},{"Start":"11:19.000 ","End":"11:23.170","Text":"we know it\u0027s a saddle and equals 0 is the dreaded case where we don\u0027t know."},{"Start":"11:23.170 ","End":"11:30.320","Text":"Let\u0027s compute the discriminant in general and then at our points 0,0 and 1,1."},{"Start":"11:31.200 ","End":"11:37.420","Text":"Here I just copy pasted the second-order derivatives."},{"Start":"11:37.420 ","End":"11:42.880","Text":"Delta or D will be equal to f_xx, f_yy,"},{"Start":"11:42.880 ","End":"11:52.360","Text":"it\u0027s 6x times 6y minus f_xy squared,"},{"Start":"11:52.360 ","End":"11:55.360","Text":"which is minus 3 squared, in other words,"},{"Start":"11:55.360 ","End":"12:02.455","Text":"it\u0027s equal to 36xy minus 9."},{"Start":"12:02.455 ","End":"12:08.005","Text":"Maybe I\u0027ll write this as Delta of x and y for emphasis."},{"Start":"12:08.005 ","End":"12:11.905","Text":"Now we\u0027re going to evaluate it at each of the critical points,"},{"Start":"12:11.905 ","End":"12:18.655","Text":"I need Delta of 0,0 and I need Delta of 1,1."},{"Start":"12:18.655 ","End":"12:25.075","Text":"Substituting 0,0 in this expression here,"},{"Start":"12:25.075 ","End":"12:28.990","Text":"we get simply minus 9,"},{"Start":"12:28.990 ","End":"12:31.510","Text":"obviously if x and y are 0."},{"Start":"12:31.510 ","End":"12:38.605","Text":"Delta at the point 1,1 is going to be 36 times 1 times 1,"},{"Start":"12:38.605 ","End":"12:43.250","Text":"minus 9, which is going to be 27."},{"Start":"12:44.550 ","End":"12:49.525","Text":"Now we\u0027ll need to start classifying them,"},{"Start":"12:49.525 ","End":"12:51.220","Text":"maybe I\u0027ll make that as a separate step."},{"Start":"12:51.220 ","End":"12:53.995","Text":"Sometimes I can combine it with Step 2."},{"Start":"12:53.995 ","End":"13:01.575","Text":"Let\u0027s say Step 3 is to classify the critical points."},{"Start":"13:01.575 ","End":"13:08.010","Text":"For 0,0, what I have is that Delta,"},{"Start":"13:08.010 ","End":"13:11.260","Text":"which is minus 9,"},{"Start":"13:11.260 ","End":"13:17.380","Text":"I have that Delta is less than 0 and if it\u0027s less than 0,"},{"Start":"13:17.380 ","End":"13:18.970","Text":"I look in here and I see,"},{"Start":"13:18.970 ","End":"13:28.805","Text":"I\u0027m in Case 3 and so immediately I say that 0,0 is the saddle point,"},{"Start":"13:28.805 ","End":"13:33.465","Text":"and I\u0027ll draw a picture in a moment and explain what a saddle point is,"},{"Start":"13:33.465 ","End":"13:36.480","Text":"it\u0027s just a concept for the moment."},{"Start":"13:36.480 ","End":"13:40.399","Text":"Next I go to the critical point 1,1,"},{"Start":"13:40.399 ","End":"13:44.155","Text":"and I see that Delta is 27,"},{"Start":"13:44.155 ","End":"13:49.300","Text":"but what\u0027s important is that it\u0027s bigger than 0, and if it\u0027s bigger than 0,"},{"Start":"13:49.300 ","End":"13:58.540","Text":"then we have to also look at f_xx."},{"Start":"13:58.540 ","End":"14:02.350","Text":"F_xx at this point is,"},{"Start":"14:02.350 ","End":"14:03.910","Text":"and I have here f_xx,"},{"Start":"14:03.910 ","End":"14:09.085","Text":"6x and if it\u0027s 6x,"},{"Start":"14:09.085 ","End":"14:11.410","Text":"then it\u0027s 6 times 1,"},{"Start":"14:11.410 ","End":"14:17.780","Text":"which is 6, and that is bigger than 0."},{"Start":"14:18.120 ","End":"14:23.109","Text":"We are in case number 1,"},{"Start":"14:23.109 ","End":"14:26.110","Text":"because Delta is bigger than 0,"},{"Start":"14:26.110 ","End":"14:27.625","Text":"D here they call it,"},{"Start":"14:27.625 ","End":"14:30.250","Text":"and f_xx is bigger than 0,"},{"Start":"14:30.250 ","End":"14:32.935","Text":"so we know that it\u0027s a minimum."},{"Start":"14:32.935 ","End":"14:39.940","Text":"This tells us that 1,1 is a minimum."},{"Start":"14:39.940 ","End":"14:47.695","Text":"That\u0027s basically it, note that if I have D or Delta positive,"},{"Start":"14:47.695 ","End":"14:50.620","Text":"although I don\u0027t know if it\u0027s a minimum or maximum,"},{"Start":"14:50.620 ","End":"14:56.425","Text":"I know it\u0027s one of them and there is a common term for minimum and maximum,"},{"Start":"14:56.425 ","End":"14:58.480","Text":"and that is the word extremum."},{"Start":"14:58.480 ","End":"15:03.220","Text":"In other words, if Delta is bigger than 0,"},{"Start":"15:03.220 ","End":"15:07.420","Text":"then I know that I have an extremum."},{"Start":"15:07.420 ","End":"15:17.320","Text":"We have to also use f_xx at the point and decide according to this."},{"Start":"15:17.320 ","End":"15:19.825","Text":"If this is bigger than 0,"},{"Start":"15:19.825 ","End":"15:24.745","Text":"then we have a minimum and if this is less than 0,"},{"Start":"15:24.745 ","End":"15:27.865","Text":"then we have a maximum."},{"Start":"15:27.865 ","End":"15:31.375","Text":"Now to properly answer the question that was originally asked,"},{"Start":"15:31.375 ","End":"15:36.475","Text":"the critical points are 0,0, and 1,1."},{"Start":"15:36.475 ","End":"15:38.395","Text":"As for the classification,"},{"Start":"15:38.395 ","End":"15:45.160","Text":"this one is a saddle point and this one is a minimum."},{"Start":"15:45.160 ","End":"15:50.500","Text":"Now I\u0027ll talk about what a saddle point is intuitively from"},{"Start":"15:50.500 ","End":"15:56.410","Text":"a geometric point of view and that will conclude this clip."},{"Start":"15:56.410 ","End":"16:00.715","Text":"First of all, here\u0027s the idea of what it looks like,"},{"Start":"16:00.715 ","End":"16:05.920","Text":"this point here is a saddle point and it should actually think of a saddle of a horse."},{"Start":"16:05.920 ","End":"16:09.775","Text":"Notice that if I go this way,"},{"Start":"16:09.775 ","End":"16:12.070","Text":"let me just try and trace that,"},{"Start":"16:12.070 ","End":"16:14.270","Text":"if I go along here,"},{"Start":"16:15.570 ","End":"16:18.280","Text":"then as far as this goes,"},{"Start":"16:18.280 ","End":"16:20.500","Text":"it\u0027s like a maximum along this curve,"},{"Start":"16:20.500 ","End":"16:22.165","Text":"it goes up and then down."},{"Start":"16:22.165 ","End":"16:24.760","Text":"But if I go the other way,"},{"Start":"16:24.760 ","End":"16:29.620","Text":"let\u0027s say I\u0027m following this line here,"},{"Start":"16:29.620 ","End":"16:33.080","Text":"then I\u0027m going down,"},{"Start":"16:33.080 ","End":"16:36.060","Text":"then up, and this is the lowest point."},{"Start":"16:36.870 ","End":"16:43.955","Text":"This is not an extremum not a maximum or minimum in any vicinity,"},{"Start":"16:43.955 ","End":"16:46.845","Text":"neighborhood of this point,"},{"Start":"16:46.845 ","End":"16:51.080","Text":"we see that there are lower points if I follow the green line and higher points if I"},{"Start":"16:51.080 ","End":"16:56.165","Text":"follow the blue line and yet it is a critical point."},{"Start":"16:56.165 ","End":"17:01.910","Text":"Here I\u0027ve drawn the tangent lines in and they\u0027re both parallel."},{"Start":"17:01.910 ","End":"17:06.470","Text":"In other words, this cross here is actually parallel to the x-y plane,"},{"Start":"17:06.470 ","End":"17:08.595","Text":"and that\u0027s what makes it a critical point."},{"Start":"17:08.595 ","End":"17:11.134","Text":"At this point it\u0027s flat,"},{"Start":"17:11.134 ","End":"17:16.170","Text":"but it goes down and up depending on which way you go and that\u0027s a saddle point."},{"Start":"17:16.170 ","End":"17:18.380","Text":"There\u0027s no point saying anymore,"},{"Start":"17:18.380 ","End":"17:21.200","Text":"picture says most of it."},{"Start":"17:21.200 ","End":"17:24.360","Text":"That\u0027s it with this clip."}],"ID":9015},{"Watched":false,"Name":"Exercise 1","Duration":"10m 4s","ChapterTopicVideoID":8735,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8735.jpeg","UploadDate":"2017-02-13T05:58:01.9630000","DurationForVideoObject":"PT10M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:01.060 ","End":"00:09.060","Text":"Here we have 1 of these critical points problems with the function of 2 variables."},{"Start":"00:09.060 ","End":"00:12.630","Text":"I\u0027m going to bring in a summary of the technique,"},{"Start":"00:12.630 ","End":"00:14.220","Text":"I won\u0027t do this in every exercise,"},{"Start":"00:14.220 ","End":"00:15.780","Text":"but I\u0027ll do it here."},{"Start":"00:15.780 ","End":"00:17.400","Text":"I\u0027m not going to read it all out,"},{"Start":"00:17.400 ","End":"00:19.625","Text":"but I\u0027ll just give you the basic steps,"},{"Start":"00:19.625 ","End":"00:21.720","Text":"and actually I\u0027ll start from the middle."},{"Start":"00:21.720 ","End":"00:25.140","Text":"We look for critical points first of all,"},{"Start":"00:25.140 ","End":"00:26.625","Text":"and it says here,"},{"Start":"00:26.625 ","End":"00:31.320","Text":"now critical point of a function,"},{"Start":"00:31.320 ","End":"00:33.060","Text":"this function f of x,"},{"Start":"00:33.060 ","End":"00:40.770","Text":"y is such that the 2 partial derivatives with respect to x and with respect to y are 0,"},{"Start":"00:40.770 ","End":"00:42.450","Text":"and if that hold for a point a,"},{"Start":"00:42.450 ","End":"00:44.955","Text":"b, then a, b is a critical point."},{"Start":"00:44.955 ","End":"00:46.800","Text":"That\u0027s going to be our first step,"},{"Start":"00:46.800 ","End":"00:47.885","Text":"there might not be any,"},{"Start":"00:47.885 ","End":"00:51.859","Text":"and there might be more than 1, all is possible."},{"Start":"00:51.859 ","End":"00:55.339","Text":"After we found our point a, b,"},{"Start":"00:55.339 ","End":"00:59.840","Text":"or before, we compute the,"},{"Start":"00:59.840 ","End":"01:01.955","Text":"it\u0027s called the discriminant,"},{"Start":"01:01.955 ","End":"01:05.280","Text":"which is this funny expression,"},{"Start":"01:07.160 ","End":"01:11.350","Text":"and that gives us a function of x and y in general."},{"Start":"01:11.350 ","End":"01:16.775","Text":"Then what we do is we substitute a, b in that,"},{"Start":"01:16.775 ","End":"01:21.355","Text":"we compute D for each critical point,"},{"Start":"01:21.355 ","End":"01:24.590","Text":"and then we just follow the table."},{"Start":"01:24.590 ","End":"01:30.170","Text":"If it\u0027s bigger than 0,"},{"Start":"01:30.170 ","End":"01:36.320","Text":"then there\u0027s still 2 sub cases depending on this f_xx and so on,"},{"Start":"01:36.320 ","End":"01:41.225","Text":"and so on, it tells us each case whether we have a minimum,"},{"Start":"01:41.225 ","End":"01:44.615","Text":"maximum, saddle or don\u0027t know."},{"Start":"01:44.615 ","End":"01:47.540","Text":"That\u0027s 4 possibilities."},{"Start":"01:47.540 ","End":"01:50.350","Text":"Let\u0027s get started,"},{"Start":"01:50.350 ","End":"01:53.760","Text":"let\u0027s do the critical points first."},{"Start":"01:53.760 ","End":"01:58.335","Text":"For that we\u0027ll need the partial derivatives so f with respect to"},{"Start":"01:58.335 ","End":"02:04.335","Text":"x is going to be 16x squared,"},{"Start":"02:04.335 ","End":"02:08.850","Text":"and then y is a constant,"},{"Start":"02:08.850 ","End":"02:12.190","Text":"so it\u0027s just 12y,"},{"Start":"02:12.200 ","End":"02:15.885","Text":"and this gives nothing,"},{"Start":"02:15.885 ","End":"02:21.960","Text":"and minus 18x gives us minus 18."},{"Start":"02:22.010 ","End":"02:28.349","Text":"Derivative with respect to y will be, this is nothing,"},{"Start":"02:28.349 ","End":"02:32.340","Text":"this gives us 12x,"},{"Start":"02:32.340 ","End":"02:38.070","Text":"this gives us 6y,"},{"Start":"02:38.070 ","End":"02:41.814","Text":"and that\u0027s it, this doesn\u0027t give us anything."},{"Start":"02:41.814 ","End":"02:45.430","Text":"Now we want both of these to equal 0, in fact,"},{"Start":"02:45.430 ","End":"02:48.860","Text":"we get 2 equations in 2 unknowns,"},{"Start":"02:48.860 ","End":"02:53.480","Text":"x and y, that this is 0 and this is 0."},{"Start":"02:53.480 ","End":"02:58.265","Text":"Silly me, 3 times 8 is actually 24,"},{"Start":"02:58.265 ","End":"03:03.240","Text":"that\u0027s better, I must learn my multiplication tables."},{"Start":"03:03.240 ","End":"03:07.385","Text":"From the second equation, which is simpler,"},{"Start":"03:07.385 ","End":"03:12.230","Text":"I can get if I bring the x over to the other side,"},{"Start":"03:12.230 ","End":"03:14.825","Text":"that\u0027s minus 12x and divide by 6."},{"Start":"03:14.825 ","End":"03:20.375","Text":"From here, we can get that y is equal to minus 2x,"},{"Start":"03:20.375 ","End":"03:29.100","Text":"and then we can substitute in the first equation and get 24x squared."},{"Start":"03:29.200 ","End":"03:36.010","Text":"Now, 12 times minus 2x is minus 24x,"},{"Start":"03:36.010 ","End":"03:40.960","Text":"and then minus 18 equals 0."},{"Start":"03:40.960 ","End":"03:44.425","Text":"This looks to me like it divides by 6,"},{"Start":"03:44.425 ","End":"03:54.320","Text":"and so we get 4x squared minus 4x minus 3 equals 0."},{"Start":"03:54.810 ","End":"03:59.440","Text":"I\u0027m not going to spend time solving a quadratic equation,"},{"Start":"03:59.440 ","End":"04:01.074","Text":"I\u0027m going to give you the answers,"},{"Start":"04:01.074 ","End":"04:07.575","Text":"that x equals either minus 1/2 or 3 over 2."},{"Start":"04:07.575 ","End":"04:17.640","Text":"Now, each x has a corresponding y so our critical points will be minus 1/2,"},{"Start":"04:17.640 ","End":"04:20.669","Text":"that\u0027s for x, and y is minus 2x,"},{"Start":"04:20.669 ","End":"04:24.015","Text":"so minus 2 times minus 1/2 is 1."},{"Start":"04:24.015 ","End":"04:29.790","Text":"The other critical point would be from 3 over 2 for x,"},{"Start":"04:29.790 ","End":"04:36.035","Text":"and multiply that by minus 2 so we get minus 3."},{"Start":"04:36.035 ","End":"04:38.225","Text":"These are like the a, b here,"},{"Start":"04:38.225 ","End":"04:42.784","Text":"and these are my critical points,"},{"Start":"04:42.784 ","End":"04:46.790","Text":"and that answers the first part of the question,"},{"Start":"04:46.790 ","End":"04:48.745","Text":"so I\u0027ll highlight this."},{"Start":"04:48.745 ","End":"04:52.170","Text":"Here\u0027s 1 and here\u0027s another,"},{"Start":"04:52.170 ","End":"04:54.225","Text":"these are the critical points."},{"Start":"04:54.225 ","End":"04:57.240","Text":"Now we have to classify them,"},{"Start":"04:57.240 ","End":"05:03.960","Text":"and for that I need to compute this and I\u0027ll need the 3 second-order partial derivative,"},{"Start":"05:03.960 ","End":"05:06.300","Text":"let\u0027s see if I can squeeze them in."},{"Start":"05:06.300 ","End":"05:11.085","Text":"I can get f_xx equals,"},{"Start":"05:11.085 ","End":"05:17.265","Text":"f_xy equals, and f_yy equals."},{"Start":"05:17.265 ","End":"05:20.230","Text":"We\u0027ll move this up a bit."},{"Start":"05:20.600 ","End":"05:28.775","Text":"F_xx means I take this and differentiate it with respect to x. Y is a constant,"},{"Start":"05:28.775 ","End":"05:34.735","Text":"so is minus 18 so all I get is 2 times 24 is 48x."},{"Start":"05:34.735 ","End":"05:44.590","Text":"F_xy, I can differentiate this with respect to y or this with respect to x,"},{"Start":"05:44.590 ","End":"05:46.820","Text":"it doesn\u0027t really matter in both cases,"},{"Start":"05:46.820 ","End":"05:50.270","Text":"I\u0027ll get 12, the constant."},{"Start":"05:50.270 ","End":"05:53.335","Text":"F_yy, I need to take this 1,"},{"Start":"05:53.335 ","End":"05:55.160","Text":"differentiate with respect to y,"},{"Start":"05:55.160 ","End":"05:59.855","Text":"just get 6, 2 of them are even constants."},{"Start":"05:59.855 ","End":"06:03.930","Text":"Now I also need this D,"},{"Start":"06:04.130 ","End":"06:07.650","Text":"so I\u0027ll just call it D,"},{"Start":"06:07.650 ","End":"06:11.340","Text":"I won\u0027t put D of x and y, D is,"},{"Start":"06:11.340 ","End":"06:17.460","Text":"this times this minus this 1 squared, that\u0027s what it says."},{"Start":"06:17.460 ","End":"06:27.700","Text":"It\u0027s 48x times 6 minus 12 squared."},{"Start":"06:28.010 ","End":"06:33.210","Text":"This comes out to be, let\u0027s see,"},{"Start":"06:33.210 ","End":"06:37.365","Text":"6 times 48 is"},{"Start":"06:37.365 ","End":"06:46.770","Text":"288x minus 144,"},{"Start":"06:46.770 ","End":"06:50.015","Text":"that\u0027s our D. Now,"},{"Start":"06:50.015 ","End":"06:53.390","Text":"we need to compute the value of D at the critical points,"},{"Start":"06:53.390 ","End":"06:54.860","Text":"this is D of x and y,"},{"Start":"06:54.860 ","End":"06:57.810","Text":"I just don\u0027t always write the x, y."},{"Start":"06:59.210 ","End":"07:03.990","Text":"D at a, b,"},{"Start":"07:04.010 ","End":"07:08.669","Text":"well, let me just do each 1 separately."},{"Start":"07:08.669 ","End":"07:12.525","Text":"The first 1 minus 1/2,"},{"Start":"07:12.525 ","End":"07:16.590","Text":"1, let\u0027s see what that comes out to be."},{"Start":"07:16.590 ","End":"07:23.310","Text":"By the way, I could write this,"},{"Start":"07:23.310 ","End":"07:25.600","Text":"I should have taken 144 out,"},{"Start":"07:25.600 ","End":"07:28.600","Text":"that\u0027s a positive number, 2x minus 1."},{"Start":"07:28.600 ","End":"07:30.610","Text":"This is the important bit,"},{"Start":"07:30.610 ","End":"07:32.890","Text":"I\u0027ll write it as 144 times,"},{"Start":"07:32.890 ","End":"07:35.215","Text":"now what\u0027s 2x minus 1?"},{"Start":"07:35.215 ","End":"07:41.105","Text":"2x is minus 1 and minus 1 so"},{"Start":"07:41.105 ","End":"07:49.110","Text":"this is minus 2 and 144 times minus 2 is negative."},{"Start":"07:49.810 ","End":"07:54.450","Text":"We are in case 3."},{"Start":"07:54.890 ","End":"07:59.955","Text":"This negative case 3 means that this"},{"Start":"07:59.955 ","End":"08:08.039","Text":"is a saddle point."},{"Start":"08:08.039 ","End":"08:13.335","Text":"The saddle point is the minus 1/2, 1,"},{"Start":"08:13.335 ","End":"08:19.935","Text":"so maybe I\u0027ll just highlight this 1 here is a saddle."},{"Start":"08:19.935 ","End":"08:22.530","Text":"Now let\u0027s check the other 1."},{"Start":"08:22.530 ","End":"08:33.095","Text":"The other 1 D at 3 over 2, minus 3, 144."},{"Start":"08:33.095 ","End":"08:39.220","Text":"We needed 2x minus 1,"},{"Start":"08:39.220 ","End":"08:40.950","Text":"doesn\u0027t relate to y at all,"},{"Start":"08:40.950 ","End":"08:47.310","Text":"2x minus 1 is 3 minus 1 is 2."},{"Start":"08:47.310 ","End":"08:51.665","Text":"144 times 2, all I care about is that it\u0027s positive."},{"Start":"08:51.665 ","End":"08:57.710","Text":"I can\u0027t conclude yet because it depends on f_xx,"},{"Start":"08:57.710 ","End":"09:08.910","Text":"so I need to check what is f_xx at 3 over 2,"},{"Start":"09:09.820 ","End":"09:15.965","Text":"minus 3, and f_xx is this."},{"Start":"09:15.965 ","End":"09:22.439","Text":"This is 48 times 3 over 2,"},{"Start":"09:23.360 ","End":"09:28.470","Text":"and then I have to actually multiply it out to say it\u0027s positive."},{"Start":"09:29.780 ","End":"09:35.360","Text":"We are in case 1 where both positive,"},{"Start":"09:35.360 ","End":"09:39.395","Text":"then we have a minimum."},{"Start":"09:39.395 ","End":"09:45.345","Text":"This gives us a minimum,"},{"Start":"09:45.345 ","End":"09:51.255","Text":"and I\u0027ll just highlight that the point 3 over 2,"},{"Start":"09:51.255 ","End":"09:55.290","Text":"minus 3 is a minimum point."},{"Start":"09:55.290 ","End":"09:57.585","Text":"We\u0027ve got the 2 critical points,"},{"Start":"09:57.585 ","End":"09:58.850","Text":"the first is a saddle,"},{"Start":"09:58.850 ","End":"10:01.220","Text":"the second is a minimum."},{"Start":"10:01.220 ","End":"10:04.440","Text":"We are done."}],"ID":9016},{"Watched":false,"Name":"Exercise 2","Duration":"7m 29s","ChapterTopicVideoID":8736,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8736.jpeg","UploadDate":"2017-02-13T06:00:12.4630000","DurationForVideoObject":"PT7M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.140 ","End":"00:04.350","Text":"In this exercise, we have to find the critical points"},{"Start":"00:04.350 ","End":"00:08.670","Text":"of this function of 2 variables defined this."},{"Start":"00:08.670 ","End":"00:11.970","Text":"When we found them to classify them as maximum,"},{"Start":"00:11.970 ","End":"00:14.925","Text":"minimum, or saddle points."},{"Start":"00:14.925 ","End":"00:17.340","Text":"We don\u0027t know how many critical points there are."},{"Start":"00:17.340 ","End":"00:20.190","Text":"There could be none and there could be many."},{"Start":"00:20.190 ","End":"00:26.290","Text":"I\u0027m going to attach a summary of the theory here."},{"Start":"00:26.570 ","End":"00:31.515","Text":"I\u0027ll just keep it there and use it for reference if we need it."},{"Start":"00:31.515 ","End":"00:34.910","Text":"Now, one of the things you want to do is"},{"Start":"00:34.910 ","End":"00:38.060","Text":"calculate the partial derivatives up to second-order."},{"Start":"00:38.060 ","End":"00:41.045","Text":"The function itself is here."},{"Start":"00:41.045 ","End":"00:46.685","Text":"Let\u0027s compute first-order first with respect to x,"},{"Start":"00:46.685 ","End":"00:50.630","Text":"which will be 3x squared,"},{"Start":"00:50.630 ","End":"00:56.350","Text":"nothing here, minus 3 here and nothing in all the rest."},{"Start":"00:56.350 ","End":"01:08.155","Text":"Then f with respect to y will be 3y squared and minus 12."},{"Start":"01:08.155 ","End":"01:10.250","Text":"That\u0027s the first order."},{"Start":"01:10.250 ","End":"01:12.690","Text":"Now second-order."},{"Start":"01:12.740 ","End":"01:18.375","Text":"There\u0027s 3 of them with respect to x twice."},{"Start":"01:18.375 ","End":"01:20.950","Text":"That will be the derivative of this,"},{"Start":"01:20.950 ","End":"01:27.885","Text":"which is 6x with respect to x y, the mixed 1."},{"Start":"01:27.885 ","End":"01:30.765","Text":"It\u0027s actually the same as f y x."},{"Start":"01:30.765 ","End":"01:34.380","Text":"You would think they might have been altogether 4,"},{"Start":"01:34.380 ","End":"01:37.605","Text":"but x y and y x come out the same."},{"Start":"01:37.605 ","End":"01:42.720","Text":"As you can see if I take this and differentiate it with respect to x, I get 0."},{"Start":"01:42.720 ","End":"01:46.615","Text":"If I differentiate this with respect to y, I also get 0."},{"Start":"01:46.615 ","End":"01:49.360","Text":"The last 1 is f y y,"},{"Start":"01:49.360 ","End":"01:54.470","Text":"and this is equal to 6y."},{"Start":"01:54.470 ","End":"02:00.275","Text":"Now the first thing we want to do is find the critical points,"},{"Start":"02:00.275 ","End":"02:05.885","Text":"which means that both of these are equal to 0, fx, and fy."},{"Start":"02:05.885 ","End":"02:10.230","Text":"We get a system 3x squared minus"},{"Start":"02:10.230 ","End":"02:20.760","Text":"3 equals 0 and 3y squared minus 12 equals 0."},{"Start":"02:21.110 ","End":"02:25.965","Text":"Now, this gives me, let\u0027s see,"},{"Start":"02:25.965 ","End":"02:28.170","Text":"3x squared equals 3,"},{"Start":"02:28.170 ","End":"02:33.255","Text":"x squared equals 1. x is plus or minus 1."},{"Start":"02:33.255 ","End":"02:36.665","Text":"Now, from here, y squared is 4,"},{"Start":"02:36.665 ","End":"02:40.475","Text":"y is plus or minus 2."},{"Start":"02:40.475 ","End":"02:44.510","Text":"This actually gives us 4 critical points."},{"Start":"02:44.510 ","End":"02:50.455","Text":"Let me just write the words critical points."},{"Start":"02:50.455 ","End":"02:53.540","Text":"This is going to answer the first part of the question."},{"Start":"02:53.540 ","End":"02:55.250","Text":"We have any 1 of these with any 1 of these,"},{"Start":"02:55.250 ","End":"03:04.580","Text":"so we have 1,1 1,2, sorry."},{"Start":"03:04.580 ","End":"03:09.440","Text":"It\u0027s 1,2. 1, minus 2,"},{"Start":"03:09.440 ","End":"03:13.225","Text":"and then minus 1,2,"},{"Start":"03:13.225 ","End":"03:19.660","Text":"and minus 1, minus 2, 4 of them."},{"Start":"03:19.660 ","End":"03:22.355","Text":"Now at these critical points,"},{"Start":"03:22.355 ","End":"03:31.505","Text":"we need to figure out which kind of critical point each of these is."},{"Start":"03:31.505 ","End":"03:40.560","Text":"We\u0027re going to use basically this expression d of x y together with fxx."},{"Start":"03:40.870 ","End":"03:43.775","Text":"Let me just scroll down a bit."},{"Start":"03:43.775 ","End":"03:50.585","Text":"There we go. I had the idea that I could make this into a table."},{"Start":"03:50.585 ","End":"03:56.670","Text":"I could see what is fxx at each of these."},{"Start":"03:56.670 ","End":"04:01.680","Text":"For example, fxx is 6x."},{"Start":"04:01.680 ","End":"04:05.030","Text":"Here and here it\u0027s going to be,"},{"Start":"04:05.030 ","End":"04:08.280","Text":"I\u0027ll use a different color, 6."},{"Start":"04:08.280 ","End":"04:11.160","Text":"Here it\u0027s going to be 6."},{"Start":"04:11.160 ","End":"04:17.010","Text":"Here it\u0027s going to be minus 6 because it\u0027s 6x here,"},{"Start":"04:17.010 ","End":"04:20.400","Text":"and here it\u0027s also going to be minus 6."},{"Start":"04:20.400 ","End":"04:23.700","Text":"I\u0027ll have a row for fxy,"},{"Start":"04:23.700 ","End":"04:26.770","Text":"and I\u0027ll have a row for fyy."},{"Start":"04:27.590 ","End":"04:39.190","Text":"Finally, I guess I\u0027ll have a row for g at these points."},{"Start":"04:39.740 ","End":"04:44.330","Text":"Continuing, I don\u0027t always do this as a table."},{"Start":"04:44.330 ","End":"04:47.345","Text":"It\u0027s just a thought, it\u0027s just a way of organizing things."},{"Start":"04:47.345 ","End":"04:49.535","Text":"fxy is 0."},{"Start":"04:49.535 ","End":"04:54.225","Text":"That\u0027s easy. 0 0 0 0."},{"Start":"04:54.225 ","End":"05:00.590","Text":"fyy second derivative with respect to y twice is 6y."},{"Start":"05:00.590 ","End":"05:01.940","Text":"We just look at the y."},{"Start":"05:01.940 ","End":"05:04.040","Text":"Here y is 2,"},{"Start":"05:04.040 ","End":"05:06.140","Text":"so this is 12."},{"Start":"05:06.140 ","End":"05:08.600","Text":"Likewise here at minus 2,"},{"Start":"05:08.600 ","End":"05:10.220","Text":"it\u0027s going to be minus 12,"},{"Start":"05:10.220 ","End":"05:13.414","Text":"and here it\u0027s going to be minus 12."},{"Start":"05:13.414 ","End":"05:20.450","Text":"D is this times this minus the middle 1 squared."},{"Start":"05:20.450 ","End":"05:23.585","Text":"Well, fxy here is 0."},{"Start":"05:23.585 ","End":"05:26.390","Text":"It just boils down to this times this,"},{"Start":"05:26.390 ","End":"05:31.480","Text":"so 6 times 12 is 72,"},{"Start":"05:31.480 ","End":"05:39.525","Text":"6 times minus 12 is minus 72 minus 6 times 12 is minus 72."},{"Start":"05:39.525 ","End":"05:40.970","Text":"Here we have minus,"},{"Start":"05:40.970 ","End":"05:44.705","Text":"minus is plus 72."},{"Start":"05:44.705 ","End":"05:51.635","Text":"Now, if this is positive,"},{"Start":"05:51.635 ","End":"05:55.325","Text":"then I still need to do another check."},{"Start":"05:55.325 ","End":"05:58.760","Text":"Let\u0027s see, let me scroll down a bit more."},{"Start":"05:58.760 ","End":"06:01.625","Text":"Yeah, let\u0027s see if we can determine what it is."},{"Start":"06:01.625 ","End":"06:04.870","Text":"Here, this is positive."},{"Start":"06:04.870 ","End":"06:09.675","Text":"That means we have to check fxx, that\u0027s also positive."},{"Start":"06:09.675 ","End":"06:12.240","Text":"We get a minimum."},{"Start":"06:12.240 ","End":"06:15.645","Text":"This 1 is a minimum."},{"Start":"06:15.645 ","End":"06:19.155","Text":"Here this is negative."},{"Start":"06:19.155 ","End":"06:21.945","Text":"That\u0027s case number 3."},{"Start":"06:21.945 ","End":"06:25.570","Text":"Then it\u0027s a saddle."},{"Start":"06:28.940 ","End":"06:33.250","Text":"Here minus 1, 2,"},{"Start":"06:33.250 ","End":"06:35.930","Text":"we have also negative,"},{"Start":"06:35.930 ","End":"06:40.425","Text":"so it\u0027s also a saddle."},{"Start":"06:40.425 ","End":"06:45.450","Text":"Here it\u0027s positive, but this is negative."},{"Start":"06:45.450 ","End":"06:52.180","Text":"We are in case 2 and we have a local maximum."},{"Start":"06:54.290 ","End":"06:57.060","Text":"We\u0027re basically done."},{"Start":"06:57.060 ","End":"07:00.655","Text":"Maybe we want to summarize it."},{"Start":"07:00.655 ","End":"07:03.260","Text":"Maybe use a bit of color."},{"Start":"07:03.260 ","End":"07:05.915","Text":"This point is a minimum,"},{"Start":"07:05.915 ","End":"07:09.530","Text":"this point is a saddle point,"},{"Start":"07:09.530 ","End":"07:13.135","Text":"this point is a saddle point,"},{"Start":"07:13.135 ","End":"07:17.030","Text":"and this point is a maximum."},{"Start":"07:17.030 ","End":"07:19.520","Text":"Max obviously means maximum,"},{"Start":"07:19.520 ","End":"07:22.265","Text":"just lazy to write it out."},{"Start":"07:22.265 ","End":"07:24.920","Text":"That answers the question."},{"Start":"07:24.920 ","End":"07:30.210","Text":"We have the critical points and we have classified them, so we\u0027re done."}],"ID":9017},{"Watched":false,"Name":"Exercise 3","Duration":"6m 44s","ChapterTopicVideoID":8737,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8737.jpeg","UploadDate":"2017-02-13T06:02:33.3700000","DurationForVideoObject":"PT6M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.405","Text":"Here we have another exercise with the function of"},{"Start":"00:03.405 ","End":"00:08.500","Text":"2 variables and we have to find the critical points and classify them."},{"Start":"00:09.740 ","End":"00:12.930","Text":"This is going to be the last time I do this,"},{"Start":"00:12.930 ","End":"00:17.159","Text":"but I\u0027m going to bring in a summary of the formulas on theory."},{"Start":"00:17.159 ","End":"00:23.290","Text":"As before, we compute the partial derivatives up to second order."},{"Start":"00:23.290 ","End":"00:27.500","Text":"We have f with respect to x,"},{"Start":"00:27.500 ","End":"00:35.730","Text":"which is 3x squared minus 3y."},{"Start":"00:35.980 ","End":"00:39.680","Text":"Then f with respect to y,"},{"Start":"00:39.680 ","End":"00:52.470","Text":"which is 3y squared minus 3x."},{"Start":"00:52.470 ","End":"00:56.940","Text":"I\u0027ll continue with the second order,"},{"Start":"00:56.940 ","End":"01:00.035","Text":"with respect to x twice."},{"Start":"01:00.035 ","End":"01:06.035","Text":"That would be this with respect to x, that\u0027s 6x."},{"Start":"01:06.035 ","End":"01:12.010","Text":"Then I have the mixed fxy, same as fyx."},{"Start":"01:12.010 ","End":"01:16.910","Text":"Either differentiate this with respect to x and get minus 3,"},{"Start":"01:16.910 ","End":"01:20.405","Text":"or this with respect to y and get minus 3."},{"Start":"01:20.405 ","End":"01:24.920","Text":"I often do that as a check to see that I\u0027m on the right track."},{"Start":"01:24.920 ","End":"01:31.790","Text":"Fyy will be this with respect to y,"},{"Start":"01:31.790 ","End":"01:36.030","Text":"which is just 6y."},{"Start":"01:36.830 ","End":"01:42.125","Text":"Next thing I want to do is find the critical points."},{"Start":"01:42.125 ","End":"01:48.350","Text":"The critical points are where f with respect to x and f with respect to y are both 0."},{"Start":"01:48.350 ","End":"01:55.370","Text":"So I take these 2 and make them into a pair of equations."},{"Start":"01:55.370 ","End":"01:56.945","Text":"So the critical points,"},{"Start":"01:56.945 ","End":"02:05.160","Text":"I will have 3x squared minus 3y equals"},{"Start":"02:05.160 ","End":"02:15.755","Text":"0 and 3y squared minus 3x equals 0."},{"Start":"02:15.755 ","End":"02:18.110","Text":"Obviously, I can divide each by 3,"},{"Start":"02:18.110 ","End":"02:21.030","Text":"why don\u0027t I just erase the 3\u0027s."},{"Start":"02:24.070 ","End":"02:28.030","Text":"Gone. That looks nicer."},{"Start":"02:28.030 ","End":"02:32.115","Text":"I just changed colors to confuse things."},{"Start":"02:32.115 ","End":"02:34.800","Text":"This gives us 2 equations,"},{"Start":"02:34.800 ","End":"02:41.870","Text":"that y equals x squared and x equals y squared,"},{"Start":"02:41.870 ","End":"02:44.395","Text":"I bring x to the other sides, which sides?"},{"Start":"02:44.395 ","End":"02:49.130","Text":"It looks strange, 2 numbers each one\u0027s the square of the other."},{"Start":"02:49.130 ","End":"02:51.635","Text":"The only solutions are 0 and 1."},{"Start":"02:51.635 ","End":"02:57.905","Text":"1 way to see this is to put y as x squared in here."},{"Start":"02:57.905 ","End":"03:02.540","Text":"So we get x equals x squared, which is x^4."},{"Start":"03:02.540 ","End":"03:09.630","Text":"Then you either get that x equals 0 or 1 equals x cubed,"},{"Start":"03:09.630 ","End":"03:11.845","Text":"so x equals 1."},{"Start":"03:11.845 ","End":"03:17.435","Text":"If x is 0, then y is 0 because y is x squared,"},{"Start":"03:17.435 ","End":"03:22.040","Text":"and if x is 1, y is x squared is also equal to 1."},{"Start":"03:22.040 ","End":"03:25.895","Text":"These are the only 2 cases where each of 2 numbers is the square of the other."},{"Start":"03:25.895 ","End":"03:29.510","Text":"In other words, our a, b, well,"},{"Start":"03:29.510 ","End":"03:32.105","Text":"our critical points, 0,"},{"Start":"03:32.105 ","End":"03:35.970","Text":"0 and 1, 1."},{"Start":"03:36.280 ","End":"03:41.090","Text":"Now, what I want to do is"},{"Start":"03:41.090 ","End":"03:49.030","Text":"to compute this expression,"},{"Start":"03:49.030 ","End":"03:53.045","Text":"this d for each of the 2 points."},{"Start":"03:53.045 ","End":"03:57.900","Text":"D together with fxx will help us to determine."},{"Start":"03:58.000 ","End":"04:00.260","Text":"I\u0027m going to erase this in a moment."},{"Start":"04:00.260 ","End":"04:04.355","Text":"Let me just record that the critical points,"},{"Start":"04:04.355 ","End":"04:08.330","Text":"which is also what we were asked to find,"},{"Start":"04:08.330 ","End":"04:14.990","Text":"0, 0 and 1, 1."},{"Start":"04:14.990 ","End":"04:16.945","Text":"I\u0027ll get rid of this."},{"Start":"04:16.945 ","End":"04:19.875","Text":"I think I\u0027ll make a table,"},{"Start":"04:19.875 ","End":"04:22.325","Text":"we take the 2 points 0,"},{"Start":"04:22.325 ","End":"04:25.315","Text":"0 and 1, 1."},{"Start":"04:25.315 ","End":"04:28.980","Text":"Let me see what each of these is equal to 0,"},{"Start":"04:28.980 ","End":"04:31.500","Text":"0, this is 0,"},{"Start":"04:31.500 ","End":"04:33.645","Text":"this is minus 3,"},{"Start":"04:33.645 ","End":"04:36.365","Text":"and this is 0, and at 1,"},{"Start":"04:36.365 ","End":"04:38.865","Text":"1 this is 6."},{"Start":"04:38.865 ","End":"04:43.405","Text":"This is still minus 3 and this is 6."},{"Start":"04:43.405 ","End":"04:47.450","Text":"Let me just move this out the way because we need another row,"},{"Start":"04:47.450 ","End":"04:51.920","Text":"which is d. I could compute it in general. They don\u0027t really need to."},{"Start":"04:51.920 ","End":"04:54.440","Text":"I just have to remember that d is"},{"Start":"04:54.440 ","End":"05:04.665","Text":"fxxfyy minus fxy squared."},{"Start":"05:04.665 ","End":"05:07.090","Text":"Put a brackets here."},{"Start":"05:07.640 ","End":"05:12.915","Text":"It\u0027s this times this minus this 1 squared."},{"Start":"05:12.915 ","End":"05:17.865","Text":"Here I get 0 times 0 minus 9."},{"Start":"05:17.865 ","End":"05:22.485","Text":"This is minus 9."},{"Start":"05:22.485 ","End":"05:27.870","Text":"Then here I get 6 times 6 minus 9."},{"Start":"05:27.870 ","End":"05:30.690","Text":"Because minus 3 squared is plus 9,"},{"Start":"05:30.690 ","End":"05:32.535","Text":"but it\u0027s a minus here."},{"Start":"05:32.535 ","End":"05:37.740","Text":"So 36 minus 9 is 27."},{"Start":"05:37.740 ","End":"05:44.230","Text":"Now for the 0, 0 I check d and it\u0027s negative."},{"Start":"05:45.490 ","End":"05:49.940","Text":"A negative takes me right away into case 3,"},{"Start":"05:49.940 ","End":"05:54.810","Text":"and that makes this a saddle point."},{"Start":"05:57.320 ","End":"06:02.420","Text":"27 is positive, so that\u0027s case 1 or 2."},{"Start":"06:02.420 ","End":"06:10.145","Text":"So I need to check fxx and I notice that this is also positive."},{"Start":"06:10.145 ","End":"06:12.949","Text":"This is positive and this is positive,"},{"Start":"06:12.949 ","End":"06:14.690","Text":"so that\u0027s case 1."},{"Start":"06:14.690 ","End":"06:20.370","Text":"So that makes this a local minimum."},{"Start":"06:21.290 ","End":"06:27.260","Text":"To summarize, these are our critical points."},{"Start":"06:27.260 ","End":"06:32.045","Text":"This 1 is a saddle point"},{"Start":"06:32.045 ","End":"06:38.620","Text":"and this 1 is a minimum point."},{"Start":"06:39.170 ","End":"06:43.930","Text":"That pretty much answers the question, so we\u0027re done."}],"ID":9018},{"Watched":false,"Name":"Exercise 4","Duration":"8m 40s","ChapterTopicVideoID":8738,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8738.jpeg","UploadDate":"2017-02-13T06:04:55.5500000","DurationForVideoObject":"PT8M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In this exercise, like several before,"},{"Start":"00:03.240 ","End":"00:05.669","Text":"we\u0027re given a function of 2 variables,"},{"Start":"00:05.669 ","End":"00:08.580","Text":"x and y, we have to find its critical points,"},{"Start":"00:08.580 ","End":"00:13.840","Text":"and then to classify each one as maximum, minimum, or saddle."},{"Start":"00:17.690 ","End":"00:23.895","Text":"One way of starting is to compute all the partial derivatives up to 2nd order."},{"Start":"00:23.895 ","End":"00:26.235","Text":"With respect to x,"},{"Start":"00:26.235 ","End":"00:32.730","Text":"we get 3 minus 3x squared."},{"Start":"00:32.730 ","End":"00:35.910","Text":"With respect to y,"},{"Start":"00:35.910 ","End":"00:45.840","Text":"we get minus 4y plus 4y cubed."},{"Start":"00:45.840 ","End":"00:53.430","Text":"Let\u0027s continue f_xx, this with respect to x, minus 6x."},{"Start":"00:53.430 ","End":"01:01.860","Text":"F_xy will be this"},{"Start":"01:01.860 ","End":"01:04.920","Text":"with respect to y or this with respect to x."},{"Start":"01:04.920 ","End":"01:07.630","Text":"Either way, it\u0027s 0."},{"Start":"01:07.640 ","End":"01:12.015","Text":"F_yy, this with respect to y."},{"Start":"01:12.015 ","End":"01:22.185","Text":"That comes out minus 4 plus 12y squared."},{"Start":"01:22.185 ","End":"01:27.030","Text":"Let\u0027s first find the critical points now."},{"Start":"01:27.030 ","End":"01:30.130","Text":"I make each of these equal to 0,"},{"Start":"01:30.130 ","End":"01:33.295","Text":"so we get a pair of equations,"},{"Start":"01:33.295 ","End":"01:35.155","Text":"but I\u0027ll cancel also."},{"Start":"01:35.155 ","End":"01:40.300","Text":"I\u0027ve got 1 minus x squared equals 0."},{"Start":"01:40.300 ","End":"01:44.440","Text":"If I divide this by 4,"},{"Start":"01:44.440 ","End":"01:52.320","Text":"I get minus y plus y cubed equals 0."},{"Start":"01:52.320 ","End":"01:59.025","Text":"Now, this one gives us that x squared equals 1,"},{"Start":"01:59.025 ","End":"02:04.695","Text":"so x equals plus or minus 1."},{"Start":"02:04.695 ","End":"02:10.350","Text":"From here, we get that y cubed minus y is 0,"},{"Start":"02:10.350 ","End":"02:18.165","Text":"so y times y squared minus 1 equals 0."},{"Start":"02:18.165 ","End":"02:24.450","Text":"Either y equals 0 or y squared minus 1 is 0,"},{"Start":"02:24.450 ","End":"02:26.580","Text":"which is plus or minus 1."},{"Start":"02:26.580 ","End":"02:28.700","Text":"two possibilities for x,"},{"Start":"02:28.700 ","End":"02:31.205","Text":"three possibilities for y."},{"Start":"02:31.205 ","End":"02:39.390","Text":"That gives us a total of six possible critical points."},{"Start":"02:39.390 ","End":"02:42.020","Text":"I\u0027ll write them."},{"Start":"02:42.410 ","End":"02:46.335","Text":"We have the critical points. Let\u0027s see."},{"Start":"02:46.335 ","End":"02:47.940","Text":"I\u0027ll begin with 1."},{"Start":"02:47.940 ","End":"02:50.350","Text":"We get 1."},{"Start":"02:50.480 ","End":"02:52.755","Text":"Then each of these,"},{"Start":"02:52.755 ","End":"02:54.990","Text":"let\u0027s say, in increasing order,"},{"Start":"02:54.990 ","End":"03:01.065","Text":"1 minus 1, 1, 0, 1, 1."},{"Start":"03:01.065 ","End":"03:03.930","Text":"Then for minus 1,"},{"Start":"03:03.930 ","End":"03:06.330","Text":"we have minus 1 minus 1,"},{"Start":"03:06.330 ","End":"03:13.620","Text":"minus 1, 0, and minus 1, 1."},{"Start":"03:13.620 ","End":"03:20.680","Text":"These are the 6 critical points."},{"Start":"03:20.680 ","End":"03:25.265","Text":"This answers the first part of the question."},{"Start":"03:25.265 ","End":"03:28.835","Text":"I could highlight these, I suppose;"},{"Start":"03:28.835 ","End":"03:32.630","Text":"1, 2, 3, 4,"},{"Start":"03:32.630 ","End":"03:35.190","Text":"5, 6 of them."},{"Start":"03:35.780 ","End":"03:42.170","Text":"What I want to do is compute the value of these for each of the points."},{"Start":"03:42.170 ","End":"03:44.420","Text":"Let\u0027s just substitute."},{"Start":"03:44.420 ","End":"03:48.845","Text":"Let\u0027s first of all substitute in minus 6x."},{"Start":"03:48.845 ","End":"03:51.064","Text":"I only have to look at the first component."},{"Start":"03:51.064 ","End":"03:54.960","Text":"I get minus 6,"},{"Start":"03:56.340 ","End":"03:59.860","Text":"minus 6, minus 6,"},{"Start":"03:59.860 ","End":"04:04.945","Text":"and then minus 6 times minus 1 is 6, 6, and 6."},{"Start":"04:04.945 ","End":"04:08.130","Text":"F_xy for all of them will be 0,"},{"Start":"04:08.130 ","End":"04:11.880","Text":"0, 0, 0, 0, 0."},{"Start":"04:11.880 ","End":"04:19.960","Text":"This, which is 12y squared minus 4,"},{"Start":"04:20.000 ","End":"04:26.485","Text":"when y is 0,"},{"Start":"04:26.485 ","End":"04:29.290","Text":"that will just give me minus 4."},{"Start":"04:29.290 ","End":"04:32.755","Text":"That\u0027s here and here."},{"Start":"04:32.755 ","End":"04:39.565","Text":"Otherwise, y squared is going to be 1 for both minus 1 and 1."},{"Start":"04:39.565 ","End":"04:43.640","Text":"I get 12 minus 4 is 8."},{"Start":"04:43.640 ","End":"04:49.105","Text":"Here\u0027s 8, 8, 8, and 8."},{"Start":"04:49.105 ","End":"04:53.264","Text":"Now, the important quantity I need,"},{"Start":"04:53.264 ","End":"04:58.200","Text":"I brought in a formula just as a reminder,"},{"Start":"04:58.200 ","End":"05:06.930","Text":"is D. D is"},{"Start":"05:06.930 ","End":"05:15.515","Text":"f_xx times f_yy minus f_xy squared."},{"Start":"05:15.515 ","End":"05:19.520","Text":"Meaning this times this minus the middle 1 squared."},{"Start":"05:19.520 ","End":"05:22.565","Text":"Now, the middle 1 squared is going to be 0 in all cases."},{"Start":"05:22.565 ","End":"05:29.075","Text":"It\u0027s just this times this minus 6 times 8 is minus 48."},{"Start":"05:29.075 ","End":"05:32.345","Text":"Here, I get plus 24."},{"Start":"05:32.345 ","End":"05:38.010","Text":"I actually don\u0027t even need the value bigger than 0 or less than 0."},{"Start":"05:38.010 ","End":"05:39.815","Text":"But I\u0027ll write the values anyway."},{"Start":"05:39.815 ","End":"05:44.240","Text":"Minus 6 times 8 is again minus 48."},{"Start":"05:44.240 ","End":"05:50.350","Text":"Here, I have 48 minus 24 and 48."},{"Start":"05:50.350 ","End":"05:57.530","Text":"Now, what\u0027s important is whether we are bigger than 0,"},{"Start":"05:57.530 ","End":"05:59.300","Text":"equal to 0, or less than 0."},{"Start":"05:59.300 ","End":"06:03.665","Text":"Here, we\u0027re less than 0, bigger than 0,"},{"Start":"06:03.665 ","End":"06:06.905","Text":"less than 0, bigger than 0,"},{"Start":"06:06.905 ","End":"06:11.790","Text":"less than 0, bigger than 0."},{"Start":"06:13.850 ","End":"06:17.120","Text":"In the case that we\u0027re bigger than 0,"},{"Start":"06:17.120 ","End":"06:20.780","Text":"we also have to look at f_xx."},{"Start":"06:20.780 ","End":"06:25.830","Text":"Here, I need to note that this is negative."},{"Start":"06:26.420 ","End":"06:29.745","Text":"We have some bigger than 0 here."},{"Start":"06:29.745 ","End":"06:37.480","Text":"This is positive and also this is positive."},{"Start":"06:37.480 ","End":"06:41.330","Text":"I don\u0027t need this anymore, but I need something else."},{"Start":"06:41.330 ","End":"06:48.090","Text":"I brought in the rules for how to decide."},{"Start":"06:48.410 ","End":"06:52.190","Text":"If I have that D is negative,"},{"Start":"06:52.190 ","End":"06:54.815","Text":"then automatically it\u0027s a saddle point."},{"Start":"06:54.815 ","End":"06:58.220","Text":"This is a saddle point."},{"Start":"06:58.220 ","End":"07:04.245","Text":"That\u0027s this 1 and so is this one,"},{"Start":"07:04.245 ","End":"07:10.440","Text":"saddle, and negative another saddle."},{"Start":"07:12.920 ","End":"07:20.310","Text":"That takes care of this, this, and this."},{"Start":"07:20.310 ","End":"07:24.120","Text":"Maybe I\u0027ll just make a little mnemonic that this one is saddle,"},{"Start":"07:24.120 ","End":"07:25.995","Text":"this one is saddle,"},{"Start":"07:25.995 ","End":"07:29.220","Text":"this one is saddle."},{"Start":"07:29.220 ","End":"07:32.220","Text":"Then if it\u0027s bigger than 0,"},{"Start":"07:32.220 ","End":"07:33.540","Text":"I have two cases."},{"Start":"07:33.540 ","End":"07:38.440","Text":"I have to look at f_xx also."},{"Start":"07:41.270 ","End":"07:45.290","Text":"Here, we see it\u0027s less than 0."},{"Start":"07:45.290 ","End":"07:46.970","Text":"When it\u0027s less than 0,"},{"Start":"07:46.970 ","End":"07:49.980","Text":"then it\u0027s a maximum."},{"Start":"07:50.060 ","End":"07:54.880","Text":"Here, we have maximum,"},{"Start":"07:55.120 ","End":"08:01.930","Text":"bigger than 0 is minimum,"},{"Start":"08:01.930 ","End":"08:07.150","Text":"and bigger than 0 is minimum."},{"Start":"08:08.450 ","End":"08:11.465","Text":"Perhaps, I should have written them over here."},{"Start":"08:11.465 ","End":"08:13.805","Text":"I\u0027ll just write them again."},{"Start":"08:13.805 ","End":"08:19.230","Text":"I\u0027ll just move them over here."},{"Start":"08:20.390 ","End":"08:25.410","Text":"What\u0027s going on here? There we are."},{"Start":"08:25.410 ","End":"08:28.380","Text":"We found the critical points,"},{"Start":"08:28.380 ","End":"08:31.500","Text":"6 of them and each of them has being classified."},{"Start":"08:31.500 ","End":"08:33.030","Text":"We have 3 saddles,"},{"Start":"08:33.030 ","End":"08:40.210","Text":"2 minima, and 1 maximum. We\u0027re done."}],"ID":9019},{"Watched":false,"Name":"Exercise 5","Duration":"10m 54s","ChapterTopicVideoID":8739,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8739.jpeg","UploadDate":"2017-02-13T06:08:40.7830000","DurationForVideoObject":"PT10M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.470","Text":"In this exercise, we have to find the critical points of"},{"Start":"00:04.470 ","End":"00:09.780","Text":"this function and then classify them to one of 3 types."},{"Start":"00:10.250 ","End":"00:14.925","Text":"Here I\u0027m going to show you a little trick that helps a bit."},{"Start":"00:14.925 ","End":"00:18.255","Text":"You can do without it but I just wanted you to be familiar"},{"Start":"00:18.255 ","End":"00:24.570","Text":"that sometimes we can separate a function into an x part times a y part."},{"Start":"00:24.570 ","End":"00:26.655","Text":"Now, in this case,"},{"Start":"00:26.655 ","End":"00:37.455","Text":"this exponent could be written as 4y minus y squared and then minus x squared."},{"Start":"00:37.455 ","End":"00:47.310","Text":"Then, using the rules of exponents that in general,"},{"Start":"00:47.310 ","End":"00:53.070","Text":"a^b plus c is a^b times a^c."},{"Start":"00:53.070 ","End":"00:56.610","Text":"We can rewrite our f of x,"},{"Start":"00:56.610 ","End":"01:02.175","Text":"y as equal to e to the power of,"},{"Start":"01:02.175 ","End":"01:04.005","Text":"I\u0027ll put the x\u0027s first,"},{"Start":"01:04.005 ","End":"01:12.180","Text":"minus x squared times e^4y minus y squared."},{"Start":"01:12.180 ","End":"01:18.735","Text":"That\u0027s the trick if you want to call it that."},{"Start":"01:18.735 ","End":"01:22.720","Text":"It will just make the computations a little bit easier."},{"Start":"01:22.720 ","End":"01:30.020","Text":"We need the partial derivatives of first-order we will get with respect to x,"},{"Start":"01:30.020 ","End":"01:32.140","Text":"y is a constant."},{"Start":"01:32.140 ","End":"01:34.170","Text":"All this is a constant."},{"Start":"01:34.170 ","End":"01:38.039","Text":"We just need to differentiate e^minus x squared"},{"Start":"01:38.039 ","End":"01:42.650","Text":"and that\u0027s e^minus x squared times inner derivative,"},{"Start":"01:42.650 ","End":"01:44.705","Text":"which is minus 2x."},{"Start":"01:44.705 ","End":"01:51.025","Text":"This constant just sticks e^4y minus y squared."},{"Start":"01:51.025 ","End":"01:54.825","Text":"With f, with respect to y, a similar thing,"},{"Start":"01:54.825 ","End":"02:00.380","Text":"we get e^minus x squared and"},{"Start":"02:00.380 ","End":"02:06.245","Text":"then the derivative of this which is just e^4y minus y squared."},{"Start":"02:06.245 ","End":"02:10.860","Text":"An inner derivative is 4 minus 2y."},{"Start":"02:12.400 ","End":"02:16.800","Text":"That\u0027s the first order"},{"Start":"02:16.800 ","End":"02:19.910","Text":"and we\u0027re going to use these in a moment to find the critical points,"},{"Start":"02:19.910 ","End":"02:27.330","Text":"but let\u0027s keep going while we\u0027re differentiating fxx is equal to,"},{"Start":"02:27.330 ","End":"02:30.440","Text":"and you differentiate this with respect to x,"},{"Start":"02:30.440 ","End":"02:32.990","Text":"now this path again is a function of y,"},{"Start":"02:32.990 ","End":"02:42.300","Text":"so I\u0027ll leave it as it is and just differentiate this using the product rule."},{"Start":"02:42.380 ","End":"02:44.850","Text":"We have this times this,"},{"Start":"02:44.850 ","End":"02:47.435","Text":"so we take the derivative of this,"},{"Start":"02:47.435 ","End":"02:49.460","Text":"which is minus 2,"},{"Start":"02:49.460 ","End":"02:58.740","Text":"and then this one as is and then we take plus this as is, because it\u0027s a minus,"},{"Start":"02:58.740 ","End":"03:05.025","Text":"I\u0027ll put minus 2x,"},{"Start":"03:05.025 ","End":"03:09.660","Text":"and then the derivative of e^minus x squared,"},{"Start":"03:09.660 ","End":"03:16.985","Text":"which is minus 2x, e^minus x squared."},{"Start":"03:16.985 ","End":"03:18.955","Text":"We had this already here."},{"Start":"03:18.955 ","End":"03:25.410","Text":"All this times the y part e^4y minus y squared,"},{"Start":"03:25.410 ","End":"03:27.530","Text":"as a constant just sticks."},{"Start":"03:27.530 ","End":"03:35.015","Text":"We could simplify this and say that this is equal to,"},{"Start":"03:35.015 ","End":"03:42.095","Text":"let\u0027s just collect, we have something e^minus x squared."},{"Start":"03:42.095 ","End":"03:45.990","Text":"We have e^4y minus y squared,"},{"Start":"03:45.990 ","End":"03:48.980","Text":"and how many e^minus x squared do we have?"},{"Start":"03:48.980 ","End":"03:53.900","Text":"We have minus 2 plus 4x squared."},{"Start":"03:53.900 ","End":"03:58.080","Text":"It\u0027s 4x squared minus 2."},{"Start":"04:00.620 ","End":"04:06.750","Text":"Now, fxy, mixed second-order partial derivative,"},{"Start":"04:06.750 ","End":"04:08.040","Text":"you can take your pick,"},{"Start":"04:08.040 ","End":"04:14.860","Text":"this with respect to y or this with respect to x. I\u0027ll take this one with respect to y."},{"Start":"04:14.860 ","End":"04:17.190","Text":"The x part stays,"},{"Start":"04:17.190 ","End":"04:23.490","Text":"which is minus 2x e^minus x squared."},{"Start":"04:23.490 ","End":"04:29.895","Text":"Now the derivative of this with respect to y is going to be"},{"Start":"04:29.895 ","End":"04:39.910","Text":"4 minus 2y for the inner derivative first and then the e^4y minus y squared."},{"Start":"04:39.910 ","End":"04:51.480","Text":"Then I want the second-order with respect to y and that will give me here according to y."},{"Start":"04:51.670 ","End":"04:57.290","Text":"We\u0027ll use the product rule again."},{"Start":"04:57.290 ","End":"05:00.400","Text":"Let me just switch the order of these 2."},{"Start":"05:00.400 ","End":"05:03.535","Text":"Yeah, push the 4 minus 2y here,"},{"Start":"05:03.535 ","End":"05:07.580","Text":"and then the e^minus x squared here."},{"Start":"05:07.580 ","End":"05:12.290","Text":"Because then I can differentiate this with respect to y."},{"Start":"05:12.290 ","End":"05:16.310","Text":"This will still be a constant, e^minus x squared."},{"Start":"05:16.310 ","End":"05:18.289","Text":"Now, here I have a product."},{"Start":"05:18.289 ","End":"05:20.270","Text":"Derivative of the first,"},{"Start":"05:20.270 ","End":"05:28.200","Text":"which is minus 2e^4y minus y squared and then plus"},{"Start":"05:28.200 ","End":"05:38.010","Text":"the 4 minus 2y as is and then the derivative of this with respect to y."},{"Start":"05:38.540 ","End":"05:41.969","Text":"Well, I\u0027m going to get another 4 minus 2y,"},{"Start":"05:41.969 ","End":"05:50.610","Text":"so it\u0027s going to be squared and then e^4y minus y squared,"},{"Start":"05:50.610 ","End":"05:55.560","Text":"which if I simplify is e^minus x squared."},{"Start":"05:55.560 ","End":"06:01.060","Text":"Let\u0027s just collect the terms."},{"Start":"06:02.500 ","End":"06:04.970","Text":"Well, I won\u0027t expand this,"},{"Start":"06:04.970 ","End":"06:10.190","Text":"but I can still write it as 4 plus 2y squared,"},{"Start":"06:10.190 ","End":"06:19.890","Text":"I\u0027ll take this one first and then minus 2 and all this e^4_y minus y squared."},{"Start":"06:20.080 ","End":"06:25.460","Text":"Okay, so we have the partial derivatives up to second order."},{"Start":"06:25.460 ","End":"06:31.080","Text":"At this point, let\u0027s look for the critical points."},{"Start":"06:31.370 ","End":"06:37.050","Text":"For critical points, I need to find points where fx and fy are both 0,"},{"Start":"06:37.050 ","End":"06:41.245","Text":"so would equals 0 here and equals 0 here."},{"Start":"06:41.245 ","End":"06:45.685","Text":"Notice that e to the power of anything is always positive at least,"},{"Start":"06:45.685 ","End":"06:48.270","Text":"point is, it\u0027s never 0."},{"Start":"06:48.270 ","End":"06:54.860","Text":"The exponents can go so we just get the set of equations,"},{"Start":"06:54.860 ","End":"07:03.500","Text":"minus 2x equals 0 and 4 minus 2y equals 0."},{"Start":"07:03.500 ","End":"07:06.620","Text":"Like I said, the exponents are not 0,"},{"Start":"07:06.620 ","End":"07:08.360","Text":"I\u0027m dividing by them."},{"Start":"07:08.360 ","End":"07:11.285","Text":"This gives us only one possibility,"},{"Start":"07:11.285 ","End":"07:16.860","Text":"x is 0 and 4 minus 4 equals 2y,"},{"Start":"07:16.860 ","End":"07:18.885","Text":"so y equals 2."},{"Start":"07:18.885 ","End":"07:21.955","Text":"I get the point 0,"},{"Start":"07:21.955 ","End":"07:24.850","Text":"2, that\u0027s the x, that\u0027s the y."},{"Start":"07:24.850 ","End":"07:27.465","Text":"This will be like a,"},{"Start":"07:27.465 ","End":"07:30.285","Text":"b in a formula."},{"Start":"07:30.285 ","End":"07:32.880","Text":"Let me erase this."},{"Start":"07:32.880 ","End":"07:37.740","Text":"In fact I don\u0027t even need this either and I\u0027ll put the formula"},{"Start":"07:37.740 ","End":"07:42.975","Text":"here that this big D called the discriminant,"},{"Start":"07:42.975 ","End":"07:44.500","Text":"though we know this formula,"},{"Start":"07:44.500 ","End":"07:45.820","Text":"I\u0027m just having it here for reference,"},{"Start":"07:45.820 ","End":"07:51.250","Text":"but I need to apply this to our point 0, 2."},{"Start":"07:51.250 ","End":"07:54.100","Text":"In other words, D of 0,"},{"Start":"07:54.100 ","End":"07:57.370","Text":"2 will be fxx at 0,"},{"Start":"07:57.370 ","End":"08:02.710","Text":"2 times fyy at 0,"},{"Start":"08:02.710 ","End":"08:10.280","Text":"2 minus fxy at 0, 2 squared."},{"Start":"08:10.280 ","End":"08:12.570","Text":"Now, I can compute each of these,"},{"Start":"08:12.570 ","End":"08:15.300","Text":"let\u0027s see, fxx at 0, 2."},{"Start":"08:15.300 ","End":"08:19.914","Text":"I can take fxx from here."},{"Start":"08:19.914 ","End":"08:24.260","Text":"Now, when x is 0,"},{"Start":"08:24.260 ","End":"08:29.900","Text":"this gives me 4 times 0 squared minus 2 will give"},{"Start":"08:29.900 ","End":"08:36.699","Text":"minus 2 and then e^minus 0 is 1,"},{"Start":"08:36.699 ","End":"08:44.250","Text":"e^4 times 2 minus 2 squared is 4 so that\u0027s e^4th."},{"Start":"08:44.250 ","End":"08:46.620","Text":"That\u0027s this one here."},{"Start":"08:46.620 ","End":"08:50.715","Text":"Next, I need this and this,"},{"Start":"08:50.715 ","End":"08:53.550","Text":"I\u0027ll just write down, I\u0027m putting 0,"},{"Start":"08:53.550 ","End":"08:57.120","Text":"2 in here, same as I did here, I put in 0,"},{"Start":"08:57.120 ","End":"09:01.750","Text":"2. x is 0, so that\u0027s 1,"},{"Start":"09:01.750 ","End":"09:06.119","Text":"so 4 minus 2 times 2,"},{"Start":"09:06.119 ","End":"09:10.575","Text":"this part is 0, 0 squared is 0 minus 2."},{"Start":"09:10.575 ","End":"09:14.205","Text":"This part is 1, this part is minus 2,"},{"Start":"09:14.205 ","End":"09:20.550","Text":"and e^4y minus y squared is 8 minus 4 is 4,"},{"Start":"09:20.550 ","End":"09:26.300","Text":"so e^4th and around brackets minus something squared."},{"Start":"09:26.300 ","End":"09:30.720","Text":"Let\u0027s see, fxy at 0, 2."},{"Start":"09:30.720 ","End":"09:33.300","Text":"Well, if x is 0,"},{"Start":"09:33.300 ","End":"09:34.930","Text":"this first part 0,"},{"Start":"09:34.930 ","End":"09:37.090","Text":"so there\u0027s no need to continue,"},{"Start":"09:37.090 ","End":"09:42.855","Text":"minus 0 squared and then just multiply 2 times 2"},{"Start":"09:42.855 ","End":"09:52.090","Text":"is 4 and then e^4th times e^4th is e^8th."},{"Start":"09:52.240 ","End":"09:55.430","Text":"We don\u0027t care about the actual value,"},{"Start":"09:55.430 ","End":"09:57.960","Text":"just that it is positive."},{"Start":"09:57.960 ","End":"09:59.800","Text":"From the theory of this thing,"},{"Start":"09:59.800 ","End":"10:02.530","Text":"when we have our discriminant positive,"},{"Start":"10:02.530 ","End":"10:03.969","Text":"we know it\u0027s an extremum,"},{"Start":"10:03.969 ","End":"10:05.470","Text":"but we don\u0027t know which kind,"},{"Start":"10:05.470 ","End":"10:07.420","Text":"maximum or minimum, not yet."},{"Start":"10:07.420 ","End":"10:13.529","Text":"Then what we do is we also look at fxx at the same point,"},{"Start":"10:13.529 ","End":"10:16.930","Text":"and this is equal to, we have it here,"},{"Start":"10:16.930 ","End":"10:23.815","Text":"minus 2e^4th and that is negative."},{"Start":"10:23.815 ","End":"10:26.515","Text":"It\u0027s negative because like I said many times,"},{"Start":"10:26.515 ","End":"10:31.510","Text":"e to the anything is positive and so according to the cases,"},{"Start":"10:31.510 ","End":"10:37.345","Text":"if this D is positive and fxx is negative,"},{"Start":"10:37.345 ","End":"10:41.060","Text":"then that means that it\u0027s a maximum."},{"Start":"10:43.400 ","End":"10:48.720","Text":"The only critical point is 0,"},{"Start":"10:48.720 ","End":"10:55.140","Text":"2, and it is a maximum and so we\u0027ve answered the question and we\u0027re done."}],"ID":9020},{"Watched":false,"Name":"Exercise 6","Duration":"8m 56s","ChapterTopicVideoID":8740,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8740.jpeg","UploadDate":"2017-02-13T06:11:06.8330000","DurationForVideoObject":"PT8M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this exercise, we have to find the critical points of"},{"Start":"00:03.780 ","End":"00:08.140","Text":"this function of 2 variables and we have to classify them."},{"Start":"00:08.150 ","End":"00:15.255","Text":"In this exercise we have to be careful because the domain is not all of the xy plane."},{"Start":"00:15.255 ","End":"00:16.710","Text":"We have a square root."},{"Start":"00:16.710 ","End":"00:23.610","Text":"We have to add a restriction that x is bigger or equal to 0."},{"Start":"00:23.610 ","End":"00:28.440","Text":"If we find any critical points on the equation and they don\u0027t satisfy this,"},{"Start":"00:28.440 ","End":"00:30.300","Text":"we have to throw them out."},{"Start":"00:30.300 ","End":"00:35.220","Text":"Also notice that for convenience of differentiation,"},{"Start":"00:35.220 ","End":"00:41.580","Text":"I can write square root of x is the same thing as x to the power of a half."},{"Start":"00:42.590 ","End":"00:48.110","Text":"As usual, we compute the partial derivatives up to 2nd order."},{"Start":"00:48.110 ","End":"00:54.260","Text":"The 1st order is f with respect to x."},{"Start":"00:54.260 ","End":"00:56.900","Text":"Now, y is a constant,"},{"Start":"00:56.900 ","End":"01:02.780","Text":"so I need the derivative of square root of x."},{"Start":"01:02.780 ","End":"01:05.780","Text":"Well, I\u0027ll use this form."},{"Start":"01:05.780 ","End":"01:13.310","Text":"It\u0027s 1.5x to the minus"},{"Start":"01:13.310 ","End":"01:16.190","Text":"a 1/2 but the y stays."},{"Start":"01:16.190 ","End":"01:19.160","Text":"I could put it in front or put it at the end. It\u0027s okay."},{"Start":"01:19.160 ","End":"01:21.410","Text":"Then this is constant."},{"Start":"01:21.410 ","End":"01:23.870","Text":"This gives me minus 1,"},{"Start":"01:23.870 ","End":"01:26.785","Text":"and this is a constant as far as x goes."},{"Start":"01:26.785 ","End":"01:30.285","Text":"Then f with respect to y."},{"Start":"01:30.285 ","End":"01:34.740","Text":"Then we just have this which is x to"},{"Start":"01:34.740 ","End":"01:45.170","Text":"the 0.5, minus 2y."},{"Start":"01:45.170 ","End":"01:48.845","Text":"Then plus 6."},{"Start":"01:48.845 ","End":"01:51.995","Text":"Let\u0027s continue to second-order."},{"Start":"01:51.995 ","End":"01:57.795","Text":"Fxx is from here,"},{"Start":"01:57.795 ","End":"02:04.800","Text":"the y stays, but here I get 0.5 times minus 0.5 is minus a 1/4."},{"Start":"02:04.800 ","End":"02:07.430","Text":"X to the, reduce this by 1,"},{"Start":"02:07.430 ","End":"02:10.910","Text":"so it\u0027s minus 1.5 or minus 3 over 2."},{"Start":"02:10.910 ","End":"02:16.710","Text":"The y stays, and that\u0027s it, fxy."},{"Start":"02:17.590 ","End":"02:21.290","Text":"Differentiate this with respect to y,"},{"Start":"02:21.290 ","End":"02:25.835","Text":"just get a 0.5 X to the minus 0.5."},{"Start":"02:25.835 ","End":"02:33.740","Text":"I\u0027d like to note that if I differentiate fy with respect to x,"},{"Start":"02:33.740 ","End":"02:35.600","Text":"I would get the same thing."},{"Start":"02:35.600 ","End":"02:38.270","Text":"That\u0027s just a check I like to do."},{"Start":"02:38.270 ","End":"02:47.584","Text":"Fyy is derivative of this with respect to y is just minus 2."},{"Start":"02:47.584 ","End":"02:49.894","Text":"These are the partial derivatives."},{"Start":"02:49.894 ","End":"02:52.550","Text":"Now let\u0027s find the critical points."},{"Start":"02:52.550 ","End":"02:58.830","Text":"The critical points are when this and this are both 0."},{"Start":"03:00.400 ","End":"03:05.550","Text":"I\u0027ll reuse this. This equals 0,"},{"Start":"03:05.550 ","End":"03:12.850","Text":"this equals 0 and look at this as 2 equations in 2 unknowns."},{"Start":"03:12.860 ","End":"03:22.385","Text":"The first 1 gives me that x to the minus 0.5y is minus 1."},{"Start":"03:22.385 ","End":"03:24.810","Text":"Let\u0027s bring the 2 over."},{"Start":"03:24.860 ","End":"03:27.830","Text":"Well, let\u0027s do a lot of things."},{"Start":"03:27.830 ","End":"03:31.385","Text":"This side I can write as y,"},{"Start":"03:31.385 ","End":"03:36.180","Text":"x to the minus 0.5 is over the square root of x."},{"Start":"03:40.400 ","End":"03:44.854","Text":"By the way, this is not going to work if x is 0."},{"Start":"03:44.854 ","End":"03:47.660","Text":"Really, I should be taking x strictly"},{"Start":"03:47.660 ","End":"03:51.305","Text":"bigger than 0 because it doesn\u0027t have a derivative at 0."},{"Start":"03:51.305 ","End":"03:57.870","Text":"We\u0027ll just take positive x. Y over the square root of x,"},{"Start":"03:57.870 ","End":"04:03.185","Text":"the 2 I\u0027ll put on the other side when I bring 1 over,"},{"Start":"04:03.185 ","End":"04:06.690","Text":"multiply by 2, so this is what we get."},{"Start":"04:07.780 ","End":"04:12.110","Text":"I see would be more useful for me to isolate y."},{"Start":"04:12.110 ","End":"04:17.645","Text":"Let\u0027s write this as y equals twice square root of x."},{"Start":"04:17.645 ","End":"04:20.940","Text":"Then I can substitute in here."},{"Start":"04:24.320 ","End":"04:28.535","Text":"This time, I\u0027ll go back to square root of x."},{"Start":"04:28.535 ","End":"04:36.155","Text":"Square root of x minus 2y is"},{"Start":"04:36.155 ","End":"04:44.970","Text":"going to be 4 square root of x plus 6 equals 0."},{"Start":"04:44.970 ","End":"04:52.870","Text":"We get from here for bringing it to the other side,"},{"Start":"04:52.870 ","End":"04:56.200","Text":"I get 3 square root of x equals 6."},{"Start":"04:56.200 ","End":"05:02.065","Text":"I get square root of x equals 2 and finally,"},{"Start":"05:02.065 ","End":"05:05.095","Text":"that x equals 4."},{"Start":"05:05.095 ","End":"05:09.940","Text":"If x is 4, I plug it in here and then I"},{"Start":"05:09.940 ","End":"05:15.400","Text":"get y equals twice root 4 which is twice 2 which is 4."},{"Start":"05:15.400 ","End":"05:19.540","Text":"This gives us the y equals 4 also."},{"Start":"05:19.540 ","End":"05:24.505","Text":"The critical point is 4,4"},{"Start":"05:24.505 ","End":"05:29.500","Text":"and that is the only critical point which makes our work easier."},{"Start":"05:29.500 ","End":"05:32.304","Text":"We only have 1 point to check."},{"Start":"05:32.304 ","End":"05:37.195","Text":"Let\u0027s evaluate these at 4,4, fxx,"},{"Start":"05:37.195 ","End":"05:42.415","Text":"at 4,4 like that first and xx,"},{"Start":"05:42.415 ","End":"05:50.700","Text":"fxy at 4,4 equals fyy at 4,4 equals."},{"Start":"05:50.700 ","End":"05:58.100","Text":"Then I want to know what is the value of this discriminant D at 4,4."},{"Start":"05:58.100 ","End":"06:01.550","Text":"I\u0027ll bring in the formula in a moment."},{"Start":"06:01.550 ","End":"06:04.520","Text":"Now, here, if I put x is 4,"},{"Start":"06:04.520 ","End":"06:11.570","Text":"y is 4, I get 4 to the power of minus 3 over 2."},{"Start":"06:11.570 ","End":"06:13.385","Text":"Well, that\u0027s a good side exercise."},{"Start":"06:13.385 ","End":"06:20.990","Text":"4 to the minus 3 over 2 is 1 over 4 to the 3 over 2,"},{"Start":"06:20.990 ","End":"06:25.910","Text":"which is 1 over the square root of 4 cubed,"},{"Start":"06:25.910 ","End":"06:29.585","Text":"which is 1 over 2 cubed, it\u0027s 1/8."},{"Start":"06:29.585 ","End":"06:38.280","Text":"This bit is 1/8 and y is 4."},{"Start":"06:39.160 ","End":"06:49.260","Text":"We get minus 1/4 with 4 gives me minus 1 times this minus 1/8."},{"Start":"06:49.260 ","End":"06:52.875","Text":"This at 4,4 will be 0.5."},{"Start":"06:52.875 ","End":"06:56.325","Text":"This is 1 over the square root of 4."},{"Start":"06:56.325 ","End":"06:58.020","Text":"I\u0027ll just show you again,"},{"Start":"06:58.020 ","End":"07:05.775","Text":"4 to the minus 0.5 is 1 over 4 to the 0.5,"},{"Start":"07:05.775 ","End":"07:12.090","Text":"1 over the square root of 4, just 0.5."},{"Start":"07:12.090 ","End":"07:21.165","Text":"This part is 0.5 times a 0.5, which is 1/4."},{"Start":"07:21.165 ","End":"07:23.835","Text":"Well, it\u0027s a constant is minus 2."},{"Start":"07:23.835 ","End":"07:28.279","Text":"Now D I\u0027ll bring in the formula."},{"Start":"07:28.279 ","End":"07:31.020","Text":"Here\u0027s the formula."},{"Start":"07:31.020 ","End":"07:36.155","Text":"It basically says this times this minus this one squared."},{"Start":"07:36.155 ","End":"07:46.160","Text":"We have minus 2 minus an 1/8 minus 1/4 squared."},{"Start":"07:46.160 ","End":"07:49.445","Text":"This time this is plus 1/4."},{"Start":"07:49.445 ","End":"07:52.865","Text":"This squared is minus the 1/16."},{"Start":"07:52.865 ","End":"07:55.010","Text":"I actually don\u0027t have to compute it."},{"Start":"07:55.010 ","End":"07:56.570","Text":"I can see it\u0027s positive already,"},{"Start":"07:56.570 ","End":"07:57.950","Text":"but if you want me to compute it,"},{"Start":"07:57.950 ","End":"08:01.010","Text":"it\u0027s 4/16 minus 1/16 is 3/16,"},{"Start":"08:01.010 ","End":"08:03.485","Text":"any event as I said, it\u0027s positive."},{"Start":"08:03.485 ","End":"08:05.990","Text":"Now when we have the D positive,"},{"Start":"08:05.990 ","End":"08:07.550","Text":"we know it\u0027s an extremum,"},{"Start":"08:07.550 ","End":"08:13.690","Text":"but we don\u0027t know whether it\u0027s a maximum or a minimum."},{"Start":"08:13.690 ","End":"08:17.130","Text":"For this, we need to check fxx."},{"Start":"08:17.210 ","End":"08:21.880","Text":"This is bigger than 0, this is less than 0."},{"Start":"08:21.880 ","End":"08:31.420","Text":"Together, this implies that this critical point is a maximum."},{"Start":"08:31.420 ","End":"08:35.450","Text":"That\u0027s right, less than 0 makes it a maximum."},{"Start":"08:36.820 ","End":"08:40.590","Text":"We have our answer."},{"Start":"08:40.590 ","End":"08:45.565","Text":"Just one last point. We need to check that this point is in our domain,"},{"Start":"08:45.565 ","End":"08:50.545","Text":"the x of it is bigger than 0."},{"Start":"08:50.545 ","End":"08:55.790","Text":"This is a legal point and yeah, we\u0027re done."}],"ID":9021},{"Watched":false,"Name":"Exercise 7","Duration":"9m 28s","ChapterTopicVideoID":8745,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8745.jpeg","UploadDate":"2017-02-13T06:25:46.3470000","DurationForVideoObject":"PT9M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this exercise, we have to find the critical points of"},{"Start":"00:03.510 ","End":"00:08.760","Text":"this function of 2 variables and then to classify them."},{"Start":"00:09.560 ","End":"00:12.450","Text":"Let\u0027s look at the domain."},{"Start":"00:12.450 ","End":"00:16.455","Text":"The domain, because of the denominator which can\u0027t be 0,"},{"Start":"00:16.455 ","End":"00:20.715","Text":"we must require that xy is not 0."},{"Start":"00:20.715 ","End":"00:22.920","Text":"That means that both of them are not 0."},{"Start":"00:22.920 ","End":"00:24.884","Text":"It means x is not 0,"},{"Start":"00:24.884 ","End":"00:28.394","Text":"and y is not 0."},{"Start":"00:28.394 ","End":"00:30.180","Text":"If you want it geometrically,"},{"Start":"00:30.180 ","End":"00:36.030","Text":"it\u0027s the plane without the axes because if we exclude 0 for x or y,"},{"Start":"00:36.030 ","End":"00:38.770","Text":"then we\u0027ve excluded the axes."},{"Start":"00:41.090 ","End":"00:43.595","Text":"We\u0027re going to have to differentiate this."},{"Start":"00:43.595 ","End":"00:46.370","Text":"Let\u0027s put it in a more convenient form."},{"Start":"00:46.370 ","End":"00:49.475","Text":"We won\u0027t have to do so much quotient rule and all that."},{"Start":"00:49.475 ","End":"00:53.640","Text":"I\u0027m going to rewrite the f of x,"},{"Start":"00:53.640 ","End":"00:57.350","Text":"y. I\u0027ll just divide each 1 out."},{"Start":"00:57.350 ","End":"01:03.905","Text":"We have x squared y squared over xy,"},{"Start":"01:03.905 ","End":"01:09.440","Text":"and then minus 8x over xy,"},{"Start":"01:09.440 ","End":"01:13.090","Text":"and then plus y over xy."},{"Start":"01:13.090 ","End":"01:16.140","Text":"If we do some cancellation,"},{"Start":"01:16.140 ","End":"01:22.500","Text":"we\u0027ll get that this is xy minus 8 over"},{"Start":"01:22.500 ","End":"01:30.040","Text":"y plus 1 over x."},{"Start":"01:30.110 ","End":"01:33.490","Text":"Let\u0027s see what the derivatives are."},{"Start":"01:33.490 ","End":"01:35.380","Text":"We need them up to 2nd order."},{"Start":"01:35.380 ","End":"01:39.820","Text":"The derivative with respect to x of xy is,"},{"Start":"01:39.820 ","End":"01:44.270","Text":"with respect to x we just get y from here,"},{"Start":"01:44.270 ","End":"01:52.690","Text":"1 over y gives minus 1 over y squared so it\u0027s plus 8 over y squared."},{"Start":"01:57.080 ","End":"02:02.080","Text":"This gives nothing because there\u0027s no x in it."},{"Start":"02:02.080 ","End":"02:07.299","Text":"1 over x, the derivative is minus 1 over x squared."},{"Start":"02:07.299 ","End":"02:11.370","Text":"As for f_y, this equals,"},{"Start":"02:11.370 ","End":"02:14.150","Text":"here I have x,"},{"Start":"02:14.150 ","End":"02:17.610","Text":"and 1 over y is minus 1 over y squared,"},{"Start":"02:17.610 ","End":"02:22.320","Text":"so we get plus 8 over y squared and this gives nothing."},{"Start":"02:22.320 ","End":"02:28.485","Text":"Let\u0027s continue to 2nd order, f_xx is,"},{"Start":"02:28.485 ","End":"02:35.080","Text":"this is constant; 1 over x squared gives us minus 2 over x cubed,"},{"Start":"02:35.080 ","End":"02:39.475","Text":"so basically we get 2 over x cubed."},{"Start":"02:39.475 ","End":"02:42.610","Text":"If you\u0027re not sure, write it as minus x to the minus"},{"Start":"02:42.610 ","End":"02:46.225","Text":"2 and you\u0027ll get 2x to the minus 3. Same thing."},{"Start":"02:46.225 ","End":"02:49.905","Text":"f_xy is equal to,"},{"Start":"02:49.905 ","End":"02:54.020","Text":"I differentiate this with respect to y, I get 1."},{"Start":"02:54.020 ","End":"02:57.770","Text":"Alternatively, you could have done f_yx and differentiated this with respect to x,"},{"Start":"02:57.770 ","End":"02:58.790","Text":"you still get 1."},{"Start":"02:58.790 ","End":"03:01.100","Text":"I just like to do that check."},{"Start":"03:01.100 ","End":"03:09.060","Text":"Then f_yy is this with respect to y."},{"Start":"03:09.060 ","End":"03:13.130","Text":"We already said that y squared becomes minus 2 over y cubed,"},{"Start":"03:13.130 ","End":"03:18.750","Text":"in this case minus 16 over y cubed."},{"Start":"03:19.520 ","End":"03:22.520","Text":"I\u0027m going to do this a little bit differently than I did"},{"Start":"03:22.520 ","End":"03:25.760","Text":"the previous exercises just to show you."},{"Start":"03:25.760 ","End":"03:28.370","Text":"Remember that we have this funny thing,"},{"Start":"03:28.370 ","End":"03:31.805","Text":"D, I put it over here."},{"Start":"03:31.805 ","End":"03:36.975","Text":"Sometimes I like to just substitute it at the critical points themselves,"},{"Start":"03:36.975 ","End":"03:40.460","Text":"but we could also do it in general for any x,"},{"Start":"03:40.460 ","End":"03:43.950","Text":"y, and later on do the substitutions."},{"Start":"03:44.780 ","End":"03:48.030","Text":"Sometimes 1 comes out shorter, sometimes the other."},{"Start":"03:48.030 ","End":"03:49.340","Text":"If it\u0027s a lot of critical points,"},{"Start":"03:49.340 ","End":"03:53.210","Text":"it\u0027s best to do it right away if you don\u0027t think there\u0027s going to be any or maybe 1."},{"Start":"03:53.210 ","End":"03:56.060","Text":"Anyway, I\u0027ll just show you what we do is we compute"},{"Start":"03:56.060 ","End":"04:00.090","Text":"this at the general point D at the point x,"},{"Start":"04:00.090 ","End":"04:01.590","Text":"y is equal to,"},{"Start":"04:01.590 ","End":"04:05.330","Text":"it\u0027s basically this times this minus this 1 squared."},{"Start":"04:05.330 ","End":"04:07.370","Text":"That\u0027s what it boils down to."},{"Start":"04:07.370 ","End":"04:16.285","Text":"It\u0027s 2 over x cubed times minus 16 over y cubed minus 1 squared."},{"Start":"04:16.285 ","End":"04:23.720","Text":"What we get is minus 32 over x cubed,"},{"Start":"04:23.720 ","End":"04:28.440","Text":"y cubed minus 1."},{"Start":"04:29.150 ","End":"04:32.570","Text":"Leave that there. Now we\u0027re going to look for critical points"},{"Start":"04:32.570 ","End":"04:35.330","Text":"and later we\u0027ll substitute the critical points in here."},{"Start":"04:35.330 ","End":"04:38.375","Text":"I don\u0027t know how many critical points there will be."},{"Start":"04:38.375 ","End":"04:41.795","Text":"There might be none, there might be many, might be only 1."},{"Start":"04:41.795 ","End":"04:43.820","Text":"You have to solve an equation."},{"Start":"04:43.820 ","End":"04:47.645","Text":"We have to let both 1st order derivatives be 0."},{"Start":"04:47.645 ","End":"04:49.985","Text":"Let this equal to 0, this equal to 0."},{"Start":"04:49.985 ","End":"04:54.080","Text":"I have 2 equations and 2 unknowns, x and y."},{"Start":"04:54.080 ","End":"04:59.810","Text":"What I\u0027ll do is from here I\u0027ll say that y equals 1 over x squared,"},{"Start":"04:59.810 ","End":"05:02.525","Text":"and then substitute it in this 1."},{"Start":"05:02.525 ","End":"05:09.675","Text":"I get that x plus 8 over,"},{"Start":"05:09.675 ","End":"05:19.120","Text":"now y squared is 1 over x squared squared, that equals 0."},{"Start":"05:21.460 ","End":"05:25.295","Text":"1 over x squared squared is 1 over x^4,"},{"Start":"05:25.295 ","End":"05:27.920","Text":"which brings the x^4 to the top."},{"Start":"05:27.920 ","End":"05:35.220","Text":"I get x plus 8x^4 equals 0."},{"Start":"05:44.230 ","End":"05:50.375","Text":"That\u0027s fine. I get x times"},{"Start":"05:50.375 ","End":"05:58.680","Text":"1 plus 8x cubed equals 0."},{"Start":"05:58.680 ","End":"06:02.255","Text":"If a product is 0, then 1 or the other is 0."},{"Start":"06:02.255 ","End":"06:06.855","Text":"Either x is 0 or this is 0,"},{"Start":"06:06.855 ","End":"06:11.440","Text":"which gives us, if this equals 0,"},{"Start":"06:11.440 ","End":"06:15.070","Text":"that x cubed is,"},{"Start":"06:15.070 ","End":"06:16.240","Text":"bring the 1 to the other side,"},{"Start":"06:16.240 ","End":"06:21.710","Text":"divide by 8, is minus 1/8."},{"Start":"06:21.710 ","End":"06:26.150","Text":"This gives us that x equals the cube root of minus 1/8,"},{"Start":"06:26.150 ","End":"06:29.050","Text":"which is minus 1/2."},{"Start":"06:29.050 ","End":"06:36.960","Text":"X equals 0 does not satisfy the domain and so this 1 is ruled out."},{"Start":"06:36.960 ","End":"06:39.355","Text":"If x is minus 1/2,"},{"Start":"06:39.355 ","End":"06:45.435","Text":"I can substitute it in this expression for y and I get y equals,"},{"Start":"06:45.435 ","End":"06:48.225","Text":"let\u0027s see, x squared is 1/4,"},{"Start":"06:48.225 ","End":"06:51.330","Text":"1 over 1/4 is 4."},{"Start":"06:51.330 ","End":"06:58.770","Text":"My critical point is going to be minus 1/2,"},{"Start":"06:58.770 ","End":"07:01.190","Text":"4. Let\u0027s write it down."},{"Start":"07:01.190 ","End":"07:03.860","Text":"That\u0027s the critical point,"},{"Start":"07:03.860 ","End":"07:06.800","Text":"and that answers the first part of the question."},{"Start":"07:06.800 ","End":"07:08.095","Text":"There is only 1."},{"Start":"07:08.095 ","End":"07:11.420","Text":"Now what we have to do is classify it."},{"Start":"07:11.420 ","End":"07:20.980","Text":"Let\u0027s see. Now we can take this D and compute what is D at the point minus 1/2, 4."},{"Start":"07:20.980 ","End":"07:30.615","Text":"Let\u0027s see. This equals minus 32 over,"},{"Start":"07:30.615 ","End":"07:34.125","Text":"x cubed is minus 1/8,"},{"Start":"07:34.125 ","End":"07:39.000","Text":"y cubed is 64,"},{"Start":"07:39.000 ","End":"07:41.610","Text":"that\u0027s 4 times 4 times 4,"},{"Start":"07:41.610 ","End":"07:47.740","Text":"and then minus 1. Let\u0027s see."},{"Start":"07:56.750 ","End":"08:00.610","Text":"I\u0027ll write this, minus goes with minus,"},{"Start":"08:00.650 ","End":"08:04.770","Text":"64/8 goes 8 times,"},{"Start":"08:04.770 ","End":"08:11.385","Text":"32/8 is 4, 4 minus 1 is 3,"},{"Start":"08:11.385 ","End":"08:14.245","Text":"and 3 is bigger than 0."},{"Start":"08:14.245 ","End":"08:17.105","Text":"When we have bigger than 0,"},{"Start":"08:17.105 ","End":"08:19.145","Text":"we know that it\u0027s an extremum,"},{"Start":"08:19.145 ","End":"08:23.450","Text":"but this alone doesn\u0027t tell us maximum or minimum."},{"Start":"08:23.450 ","End":"08:33.580","Text":"For this, I need to go to the 2nd derivative f_xx and see what that is."},{"Start":"08:35.960 ","End":"08:39.165","Text":"F_xx at the point minus 1/2,"},{"Start":"08:39.165 ","End":"08:44.295","Text":"4 is equal to 2 over x cubed."},{"Start":"08:44.295 ","End":"08:50.680","Text":"X is minus 1/2 so this denominator is minus 1/8."},{"Start":"08:51.710 ","End":"08:56.150","Text":"All together I get minus 16."},{"Start":"08:56.150 ","End":"08:58.790","Text":"I could have seen already that it\u0027s going to be negative,"},{"Start":"08:58.790 ","End":"09:01.920","Text":"which is really all that matters. It\u0027s negative."},{"Start":"09:07.060 ","End":"09:09.425","Text":"We have the point,"},{"Start":"09:09.425 ","End":"09:14.280","Text":"we have its classification that it\u0027s a minimum point,"},{"Start":"09:18.070 ","End":"09:22.245","Text":"and this and this give us the answer."},{"Start":"09:22.245 ","End":"09:27.130","Text":"Just highlight it and we are done."}],"ID":9022},{"Watched":false,"Name":"Exercise 8","Duration":"5m 2s","ChapterTopicVideoID":8746,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8746.jpeg","UploadDate":"2017-02-13T06:26:22.3230000","DurationForVideoObject":"PT5M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.230","Text":"In this exercise, we have this function of 2 variables,"},{"Start":"00:04.230 ","End":"00:05.710","Text":"e to the x cosine and y."},{"Start":"00:05.710 ","End":"00:11.140","Text":"We want to find its critical points and then to classify them."},{"Start":"00:11.720 ","End":"00:14.130","Text":"We don\u0027t know how many there\u0027s going to be,"},{"Start":"00:14.130 ","End":"00:16.410","Text":"it might be 1, it might be many, it might be none."},{"Start":"00:16.410 ","End":"00:22.155","Text":"Let\u0027s check. We\u0027ll start off with the derivatives."},{"Start":"00:22.155 ","End":"00:24.625","Text":"Let\u0027s do the first order."},{"Start":"00:24.625 ","End":"00:29.579","Text":"Derivative with respect to x. Y is a constant,"},{"Start":"00:29.579 ","End":"00:34.495","Text":"so it\u0027s just e_x and the constant sticks."},{"Start":"00:34.495 ","End":"00:38.390","Text":"Derivative with respect to y. I"},{"Start":"00:38.390 ","End":"00:42.755","Text":"keep this constant and the derivative of this is minus sine y,"},{"Start":"00:42.755 ","End":"00:47.380","Text":"so minus e_x sine y."},{"Start":"00:47.380 ","End":"00:50.360","Text":"Now second-order with respect to x,"},{"Start":"00:50.360 ","End":"00:53.060","Text":"differentiate this with respect to x."},{"Start":"00:53.060 ","End":"00:58.625","Text":"It\u0027s still e_x cosine y doesn\u0027t matter how many times you differentiate it,"},{"Start":"00:58.625 ","End":"01:01.925","Text":"a constant times e_x. That\u0027s what it\u0027ll be."},{"Start":"01:01.925 ","End":"01:05.550","Text":"The mixed f x, y."},{"Start":"01:05.770 ","End":"01:11.675","Text":"You can take or pick either this with respect to y of this with respect to x."},{"Start":"01:11.675 ","End":"01:18.150","Text":"In both cases, you\u0027re going to get minus e_x sine y."},{"Start":"01:20.690 ","End":"01:24.480","Text":"With respect to y twice,"},{"Start":"01:24.480 ","End":"01:31.445","Text":"we just have to differentiate the sine y and the rest of it stays the same."},{"Start":"01:31.445 ","End":"01:34.740","Text":"It\u0027s minus e_x"},{"Start":"01:36.400 ","End":"01:42.350","Text":"cosine y."},{"Start":"01:42.350 ","End":"01:47.410","Text":"Let\u0027s start off with finding the critical points."},{"Start":"01:47.990 ","End":"01:51.740","Text":"I want the first-order partials to both be 0."},{"Start":"01:51.740 ","End":"01:54.590","Text":"I need this to be 0 and I need this to be"},{"Start":"01:54.590 ","End":"01:58.385","Text":"0 and I\u0027ll take those 2 equations and 2 unknowns."},{"Start":"01:58.385 ","End":"02:03.100","Text":"The thing is that e_x is always positive."},{"Start":"02:03.100 ","End":"02:06.920","Text":"That\u0027s less important than the fact that it\u0027s not 0."},{"Start":"02:06.920 ","End":"02:09.920","Text":"If it\u0027s not 0, we can divide by it."},{"Start":"02:09.920 ","End":"02:15.890","Text":"We get cosine y equals 0 and minus sine y is 0."},{"Start":"02:15.890 ","End":"02:19.760","Text":"In other words, sine y is 0."},{"Start":"02:19.760 ","End":"02:23.875","Text":"Now, I claim that this has no solution."},{"Start":"02:23.875 ","End":"02:26.645","Text":"I\u0027ll even show you 2 different ways of doing it."},{"Start":"02:26.645 ","End":"02:36.865","Text":"1 way of doing it is to say that sine squared of y plus cosine squared y,"},{"Start":"02:36.865 ","End":"02:40.850","Text":"so as we can, cosecant squared is always 1."},{"Start":"02:40.960 ","End":"02:45.050","Text":"But if I substitute from here, on the other hand,"},{"Start":"02:45.050 ","End":"02:49.640","Text":"I\u0027ll get 0 squared plus 0 squared which is 0."},{"Start":"02:49.640 ","End":"02:54.615","Text":"I get 1 equals 0, that\u0027s a contradiction."},{"Start":"02:54.615 ","End":"02:56.385","Text":"There is no solution,"},{"Start":"02:56.385 ","End":"03:01.410","Text":"so that means that there are no critical points."},{"Start":"03:03.020 ","End":"03:05.600","Text":"If there\u0027s no critical points,"},{"Start":"03:05.600 ","End":"03:07.355","Text":"then I can\u0027t classify them."},{"Start":"03:07.355 ","End":"03:09.320","Text":"But it also means there\u0027s no maximum,"},{"Start":"03:09.320 ","End":"03:12.325","Text":"minimum or a saddle for this function."},{"Start":"03:12.325 ","End":"03:16.550","Text":"Now I use that trick with the sine squared plus cosine squared."},{"Start":"03:16.550 ","End":"03:19.265","Text":"Suppose you didn\u0027t see the trick, so what would you do?"},{"Start":"03:19.265 ","End":"03:23.410","Text":"You\u0027d go ahead and solve each of these."},{"Start":"03:23.410 ","End":"03:30.180","Text":"Knowing trigonometrical equations, we would get from cosine y equals 0."},{"Start":"03:31.160 ","End":"03:37.290","Text":"The general solution is that y is equal to"},{"Start":"03:37.290 ","End":"03:46.500","Text":"Pi over 2 plus kPi."},{"Start":"03:46.500 ","End":"03:51.580","Text":"Turns out that, cosine 0 at 90 and 270,"},{"Start":"03:51.580 ","End":"03:55.235","Text":"and so on, every Pi will repeat itself."},{"Start":"03:55.235 ","End":"03:59.390","Text":"Whereas the sine y equals 0 has"},{"Start":"03:59.390 ","End":"04:06.095","Text":"a general solution that y is some multiple of a 180."},{"Start":"04:06.095 ","End":"04:08.945","Text":"But I can\u0027t use the same letter k again,"},{"Start":"04:08.945 ","End":"04:10.355","Text":"could be a different case."},{"Start":"04:10.355 ","End":"04:15.710","Text":"I\u0027ll use letter l after k, l times Pi,"},{"Start":"04:15.710 ","End":"04:19.670","Text":"where k and l are integers,"},{"Start":"04:19.670 ","End":"04:22.840","Text":"positive or negative but whole."},{"Start":"04:22.840 ","End":"04:26.215","Text":"Then if I compare these 2,"},{"Start":"04:26.215 ","End":"04:28.640","Text":"because they\u0027re both equal to y,"},{"Start":"04:28.640 ","End":"04:32.820","Text":"I get that Pi over"},{"Start":"04:32.820 ","End":"04:39.440","Text":"2 plus kPi equals lPi."},{"Start":"04:39.440 ","End":"04:43.370","Text":"If I divide everything by Pi and bring the k to the other side,"},{"Start":"04:43.370 ","End":"04:48.710","Text":"I\u0027ll get l minus k equals 1.5."},{"Start":"04:48.710 ","End":"04:49.939","Text":"But they\u0027re integers."},{"Start":"04:49.939 ","End":"04:53.305","Text":"You can\u0027t subtract 2 integers and get a fraction."},{"Start":"04:53.305 ","End":"04:56.235","Text":"That\u0027s also a contradiction."},{"Start":"04:56.235 ","End":"04:59.405","Text":"Again, it shows that there is no solution to this equation,"},{"Start":"04:59.405 ","End":"05:02.879","Text":"and hence no critical points."}],"ID":9023},{"Watched":false,"Name":"Exercise 9","Duration":"5m 34s","ChapterTopicVideoID":8741,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8741.jpeg","UploadDate":"2017-02-13T06:13:02.8300000","DurationForVideoObject":"PT5M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.470","Text":"In this exercise, we\u0027re given a surface z as a function of x and y."},{"Start":"00:07.470 ","End":"00:14.385","Text":"In fact, I can even write this as f of x and y in case we need that notation."},{"Start":"00:14.385 ","End":"00:19.570","Text":"We have to find the equation of the tangent planes which are horizontal."},{"Start":"00:19.570 ","End":"00:25.160","Text":"Now it turns out that tangent plane is horizontal"},{"Start":"00:25.160 ","End":"00:33.910","Text":"exactly when the partial derivatives fx and fy are both 0."},{"Start":"00:33.910 ","End":"00:41.630","Text":"In other words, the critical points are the same as"},{"Start":"00:41.630 ","End":"00:47.120","Text":"the points where"},{"Start":"00:47.120 ","End":"00:57.215","Text":"the tangent is horizontal."},{"Start":"00:57.215 ","End":"01:02.955","Text":"You can either just take it on trust if you want a quick explanation."},{"Start":"01:02.955 ","End":"01:08.975","Text":"When we have a surface z equals f of x, y,"},{"Start":"01:08.975 ","End":"01:17.760","Text":"then we showed at some point before that a normal vector to any point,"},{"Start":"01:17.760 ","End":"01:27.580","Text":"a normal vector to the surface at a point was equal to f with respect to x,"},{"Start":"01:27.580 ","End":"01:35.855","Text":"i plus f with respect to x, j minus 1."},{"Start":"01:35.855 ","End":"01:40.195","Text":"I think we did it maybe with the other notation, and said fx,"},{"Start":"01:40.195 ","End":"01:45.314","Text":"fy minus 1 is a normal vector."},{"Start":"01:45.314 ","End":"01:48.760","Text":"Now this isn\u0027t the vertical direction,"},{"Start":"01:48.760 ","End":"01:52.595","Text":"the minus or plus doesn\u0027t make any difference in direction of z."},{"Start":"01:52.595 ","End":"01:54.500","Text":"For a horizontal plane,"},{"Start":"01:54.500 ","End":"01:56.090","Text":"it has to be strictly vertical,"},{"Start":"01:56.090 ","End":"02:00.260","Text":"which means that this has to be 0 and this has to be 0."},{"Start":"02:00.260 ","End":"02:03.095","Text":"Then it\u0027s an a direction of 0, 0, 1,"},{"Start":"02:03.095 ","End":"02:06.350","Text":"the negative z-axis or the positive z-axis,"},{"Start":"02:06.350 ","End":"02:08.795","Text":"it doesn\u0027t really matter,"},{"Start":"02:08.795 ","End":"02:12.470","Text":"and that\u0027s why we set fx is 0 and fy is 0."},{"Start":"02:12.470 ","End":"02:14.585","Text":"Let\u0027s actually do that."},{"Start":"02:14.585 ","End":"02:17.395","Text":"Now let\u0027s see here\u0027s that,"},{"Start":"02:17.395 ","End":"02:22.725","Text":"so f with respect to x is"},{"Start":"02:22.725 ","End":"02:28.740","Text":"equal to 3x squared nothing from here,"},{"Start":"02:28.740 ","End":"02:31.885","Text":"from here minus 3y."},{"Start":"02:31.885 ","End":"02:37.430","Text":"Nothing from there, and f with respect to y will"},{"Start":"02:37.430 ","End":"02:44.940","Text":"give us 3y squared minus 3x."},{"Start":"02:44.940 ","End":"02:52.620","Text":"Now we want to see this as 2 equations and 2 unknowns,"},{"Start":"02:52.620 ","End":"02:55.780","Text":"and we want each of these to be 0."},{"Start":"02:56.810 ","End":"03:03.360","Text":"Well, divide everything by 3 just get rid of the 3s."},{"Start":"03:03.360 ","End":"03:11.325","Text":"Then we have from here that both y equals x squared,"},{"Start":"03:11.325 ","End":"03:14.955","Text":"and x equals y squared."},{"Start":"03:14.955 ","End":"03:18.365","Text":"I know we\u0027ve done this in the previous exercise,"},{"Start":"03:18.365 ","End":"03:22.175","Text":"and that turned out to be only 2 solutions."},{"Start":"03:22.175 ","End":"03:25.070","Text":"They could be both 0 or both 1,"},{"Start":"03:25.070 ","End":"03:26.270","Text":"I won\u0027t get into it again."},{"Start":"03:26.270 ","End":"03:28.085","Text":"If you want to know the idea,"},{"Start":"03:28.085 ","End":"03:31.490","Text":"the idea is to let y equals x squared in this equation,"},{"Start":"03:31.490 ","End":"03:34.060","Text":"you get x equals x^4,"},{"Start":"03:34.060 ","End":"03:42.010","Text":"and then you get that x is either 0 or 1,"},{"Start":"03:42.010 ","End":"03:44.220","Text":"and similarly for y."},{"Start":"03:44.220 ","End":"03:45.780","Text":"We have 2 solutions,"},{"Start":"03:45.780 ","End":"03:47.430","Text":"we have the 0,"},{"Start":"03:47.430 ","End":"03:55.990","Text":"0 critical point, the 1, 1 critical point for x and y."},{"Start":"03:56.270 ","End":"04:05.150","Text":"These are the points of where the tangent planes or horizontal."},{"Start":"04:05.150 ","End":"04:12.200","Text":"Now, the equation of a horizontal plane is going to be z equals constant,."},{"Start":"04:12.200 ","End":"04:16.730","Text":"All I have to do is figure out the value of z for these,"},{"Start":"04:16.730 ","End":"04:20.150","Text":"so I need f of 0, 0,"},{"Start":"04:20.150 ","End":"04:23.090","Text":"that\u0027s the z for this point, what it equals 2,"},{"Start":"04:23.090 ","End":"04:24.860","Text":"and the need of f of 1,"},{"Start":"04:24.860 ","End":"04:26.750","Text":"1 is what this equals."},{"Start":"04:26.750 ","End":"04:29.360","Text":"If I put in 0, 0,"},{"Start":"04:29.360 ","End":"04:33.790","Text":"I\u0027ve got that z or f is 4,"},{"Start":"04:33.790 ","End":"04:37.130","Text":"and if I put in 1,"},{"Start":"04:37.130 ","End":"04:44.300","Text":"1, 1 plus 1 is 2 keep the pluses plus 4 is 6,"},{"Start":"04:44.300 ","End":"04:49.220","Text":"minus 3 is 3,"},{"Start":"04:49.220 ","End":"04:52.120","Text":"unless I made a mistake."},{"Start":"04:52.310 ","End":"04:59.180","Text":"So at 0, 0 the tangent plane is just z equals 4,"},{"Start":"04:59.180 ","End":"05:01.880","Text":"which is horizontal, and at 1,"},{"Start":"05:01.880 ","End":"05:06.305","Text":"1, the tangent line is z equals 3."},{"Start":"05:06.305 ","End":"05:10.100","Text":"Yeah, x and y don\u0027t appear because a plane horizontal to the x,"},{"Start":"05:10.100 ","End":"05:14.240","Text":"y plane is just z equals something."},{"Start":"05:14.240 ","End":"05:19.250","Text":"This is the solution 1 and this is solution 2."},{"Start":"05:19.250 ","End":"05:24.300","Text":"I\u0027ll just highlight them they didn\u0027t even ask at which points they occur,"},{"Start":"05:24.300 ","End":"05:25.650","Text":"but if they did,"},{"Start":"05:25.650 ","End":"05:28.950","Text":"this 1 is tangent when x,"},{"Start":"05:28.950 ","End":"05:30.570","Text":"y are 0, 0,"},{"Start":"05:30.570 ","End":"05:32.205","Text":"and this 1 at 1,1."},{"Start":"05:32.205 ","End":"05:35.380","Text":"But that\u0027s all they asked for, and so we\u0027re done."}],"ID":9024},{"Watched":false,"Name":"Exercise 10","Duration":"11m 19s","ChapterTopicVideoID":8742,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8742.jpeg","UploadDate":"2017-02-13T06:16:46.1800000","DurationForVideoObject":"PT11M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"This is an exercise about boxes."},{"Start":"00:03.840 ","End":"00:07.275","Text":"Those are 3-dimensional rectangle, to speak."},{"Start":"00:07.275 ","End":"00:12.810","Text":"Now, there\u0027s many of them who have a volume of 32 cubic centimeters,"},{"Start":"00:12.810 ","End":"00:16.860","Text":"but we want to find the 1 with least surface area;"},{"Start":"00:16.860 ","End":"00:19.170","Text":"its dimensions, length, width, and height."},{"Start":"00:19.170 ","End":"00:22.440","Text":"I\u0027ll start right away with a picture. Here we go."},{"Start":"00:22.440 ","End":"00:24.660","Text":"A box with dimensions,"},{"Start":"00:24.660 ","End":"00:30.900","Text":"width x, height no length y, height is z."},{"Start":"00:30.900 ","End":"00:35.525","Text":"The 1 that\u0027s important is the height because when we say open boxes,"},{"Start":"00:35.525 ","End":"00:38.935","Text":"we mean a cardboard box that is open."},{"Start":"00:38.935 ","End":"00:41.180","Text":"When we say surface area,"},{"Start":"00:41.180 ","End":"00:42.760","Text":"we talking about 5 sides,"},{"Start":"00:42.760 ","End":"00:46.960","Text":"there\u0027s the base and the 4 walls but without the roof."},{"Start":"00:46.960 ","End":"00:52.505","Text":"In this case, let\u0027s take a letter S for surface area."},{"Start":"00:52.505 ","End":"00:57.585","Text":"We have that the surface area is equal to,"},{"Start":"00:57.585 ","End":"01:00.975","Text":"we have the base, which is xy."},{"Start":"01:00.975 ","End":"01:09.860","Text":"Then we have this side is x times z but this 1 is the same,"},{"Start":"01:09.860 ","End":"01:11.390","Text":"so there\u0027s 2 of those."},{"Start":"01:11.390 ","End":"01:17.000","Text":"Then there\u0027s this rectangle here which is y by z."},{"Start":"01:17.000 ","End":"01:20.280","Text":"But again there\u0027s 2 of them, so it\u0027s 2yz."},{"Start":"01:21.080 ","End":"01:25.985","Text":"Now, it looks like S is a function of 3 variables, xyz."},{"Start":"01:25.985 ","End":"01:28.745","Text":"I want to eliminate 1 of the variables."},{"Start":"01:28.745 ","End":"01:31.940","Text":"I can eliminate z because its a piece of information we haven\u0027t"},{"Start":"01:31.940 ","End":"01:36.320","Text":"used that the volume is 32 cubic centimeters."},{"Start":"01:36.320 ","End":"01:41.600","Text":"V equals 32, which means the volume is"},{"Start":"01:41.600 ","End":"01:47.385","Text":"x times y times z is 32."},{"Start":"01:47.385 ","End":"01:51.530","Text":"I can express z in terms of x and y and say"},{"Start":"01:51.530 ","End":"01:58.050","Text":"that z is 32 over xy."},{"Start":"01:58.050 ","End":"02:00.590","Text":"Now, if I put this z in here,"},{"Start":"02:00.590 ","End":"02:04.355","Text":"I\u0027ll get S as a function of just x and y,"},{"Start":"02:04.355 ","End":"02:07.230","Text":"which is equal to xy"},{"Start":"02:07.230 ","End":"02:20.225","Text":"plus 2x times 32 over xy."},{"Start":"02:20.225 ","End":"02:31.275","Text":"Then 2y times also 32 over xy."},{"Start":"02:31.275 ","End":"02:33.875","Text":"I can rewrite this."},{"Start":"02:33.875 ","End":"02:36.785","Text":"S is a function of x and y,"},{"Start":"02:36.785 ","End":"02:43.450","Text":"and it\u0027s equal to xy plus now we can do a bit of canceling."},{"Start":"02:43.450 ","End":"02:46.499","Text":"All these dimensions are positive,"},{"Start":"02:46.499 ","End":"02:48.170","Text":"there\u0027s no zeros here."},{"Start":"02:48.170 ","End":"02:50.330","Text":"X cancels with x,"},{"Start":"02:50.330 ","End":"02:52.310","Text":"y cancels with y,"},{"Start":"02:52.310 ","End":"02:59.145","Text":"and 2 times 32 gives us 64 here and here."},{"Start":"02:59.145 ","End":"03:09.529","Text":"Continuing, this is 64 over y and this is 64 over x."},{"Start":"03:09.529 ","End":"03:15.640","Text":"Now, we have an extremum problem or specifically a minimum problem in 2 variables."},{"Start":"03:15.640 ","End":"03:18.860","Text":"We\u0027ll do it by finding the critical points and checking"},{"Start":"03:18.860 ","End":"03:22.250","Text":"the conditions to see whether we have maximum,"},{"Start":"03:22.250 ","End":"03:26.630","Text":"minimum or saddle and we are looking for minimum."},{"Start":"03:26.630 ","End":"03:33.725","Text":"Let\u0027s compute the first-order partial derivatives S_x and S_y."},{"Start":"03:33.725 ","End":"03:36.800","Text":"S_x, it will be from here,"},{"Start":"03:36.800 ","End":"03:38.795","Text":"we get y from here,"},{"Start":"03:38.795 ","End":"03:42.410","Text":"nothing because it\u0027s constant as far as x goes."},{"Start":"03:42.410 ","End":"03:46.370","Text":"From here, the derivative of 1 over x is minus 1 over x"},{"Start":"03:46.370 ","End":"03:52.615","Text":"squared so we get minus 64 over x squared."},{"Start":"03:52.615 ","End":"03:57.895","Text":"With respect to y, very similar here we get x,"},{"Start":"03:57.895 ","End":"04:03.560","Text":"and then it\u0027s going to be 64 over y squared."},{"Start":"04:04.100 ","End":"04:08.920","Text":"Let\u0027s do the second-order partial derivatives."},{"Start":"04:08.920 ","End":"04:11.110","Text":"I need a bit more space."},{"Start":"04:11.110 ","End":"04:12.610","Text":"There we are."},{"Start":"04:12.610 ","End":"04:21.950","Text":"Let\u0027s figure out what is S_xx, S_xy, and S_yy."},{"Start":"04:21.950 ","End":"04:24.480","Text":"What are these equal to?"},{"Start":"04:24.480 ","End":"04:28.045","Text":"For the first 1, we differentiate this with respect to x."},{"Start":"04:28.045 ","End":"04:34.310","Text":"The y goes minus 1 over x squared gives us 2 over x cubed."},{"Start":"04:34.310 ","End":"04:43.950","Text":"Basically, comes out to be plus a 128 that\u0027s twice 64 over x cubed."},{"Start":"04:43.950 ","End":"04:49.324","Text":"S_xy, this with respect to y is just 1,"},{"Start":"04:49.324 ","End":"04:53.485","Text":"which I note also is the same as this with respect to x;"},{"Start":"04:53.485 ","End":"04:57.215","Text":"2 mixed order partial derivatives are equal."},{"Start":"04:57.215 ","End":"04:59.750","Text":"S_yy will be just like with the x,"},{"Start":"04:59.750 ","End":"05:04.890","Text":"we\u0027ll get 128 over y cubed."},{"Start":"05:05.360 ","End":"05:08.195","Text":"Let\u0027s look for critical points."},{"Start":"05:08.195 ","End":"05:11.810","Text":"Critical points are when both the first-order derivatives are 0,"},{"Start":"05:11.810 ","End":"05:14.450","Text":"so I\u0027ll just put a curly brace"},{"Start":"05:14.450 ","End":"05:18.020","Text":"here and make this into an equation I won\u0027t just copy it again."},{"Start":"05:18.020 ","End":"05:21.330","Text":"This equals 0, this equals 0."},{"Start":"05:23.110 ","End":"05:30.140","Text":"From here, I get y in terms of x as 64 over x squared."},{"Start":"05:30.140 ","End":"05:32.989","Text":"Now, if I plug that into here,"},{"Start":"05:32.989 ","End":"05:42.290","Text":"I will get x minus 64 over y"},{"Start":"05:42.290 ","End":"05:47.285","Text":"squared is 64"},{"Start":"05:47.285 ","End":"05:54.070","Text":"squared over x^4."},{"Start":"05:54.070 ","End":"05:55.335","Text":"If I take this squared,"},{"Start":"05:55.335 ","End":"05:57.630","Text":"I square top and bottom,"},{"Start":"05:57.630 ","End":"06:03.660","Text":"and this is equal to 0, x minus."},{"Start":"06:03.660 ","End":"06:07.850","Text":"Now, dividing by a fraction is multiplying by the inverse fraction."},{"Start":"06:07.850 ","End":"06:18.735","Text":"It\u0027s 64 times the reverse of this x^4 over 64 squared and this is 0."},{"Start":"06:18.735 ","End":"06:22.560","Text":"Now, 64 cancels with 1 of the 64,"},{"Start":"06:22.560 ","End":"06:25.185","Text":"so it\u0027s like I crossed out the 2."},{"Start":"06:25.185 ","End":"06:27.119","Text":"To get rid of fractions,"},{"Start":"06:27.119 ","End":"06:30.435","Text":"I\u0027ll multiply both sides by 64."},{"Start":"06:30.435 ","End":"06:34.865","Text":"Just indicate that I\u0027m multiplying both sides by 64,"},{"Start":"06:34.865 ","End":"06:43.590","Text":"I get 64x minus x^4 equals 0."},{"Start":"06:43.850 ","End":"06:49.080","Text":"Then x is not 0 so I can divide by"},{"Start":"06:49.080 ","End":"06:54.140","Text":"x. I have to make sure that it\u0027s not 0. We can\u0027t have x is 0."},{"Start":"06:54.140 ","End":"06:55.640","Text":"All the dimensions are positive."},{"Start":"06:55.640 ","End":"07:01.445","Text":"I can even write that xy and z after all be positive."},{"Start":"07:01.445 ","End":"07:06.650","Text":"Then I get that 64 equals x cubed or let\u0027s just say"},{"Start":"07:06.650 ","End":"07:12.725","Text":"x cubed equals 64 after dividing by x and bringing over."},{"Start":"07:12.725 ","End":"07:16.040","Text":"X is equal to 4."},{"Start":"07:16.040 ","End":"07:18.110","Text":"Now, if you look here,"},{"Start":"07:18.110 ","End":"07:20.900","Text":"you see that y is 64 over x squared."},{"Start":"07:20.900 ","End":"07:27.125","Text":"So that gives us that y equals 64 over 4 squared,"},{"Start":"07:27.125 ","End":"07:28.940","Text":"which is 64 over 16,"},{"Start":"07:28.940 ","End":"07:31.210","Text":"y is also 4."},{"Start":"07:31.210 ","End":"07:35.295","Text":"Our critical point is the point"},{"Start":"07:35.295 ","End":"07:42.080","Text":"4,4 which just means x equals 4y equals 4;"},{"Start":"07:42.080 ","End":"07:45.005","Text":"that\u0027s the critical point."},{"Start":"07:45.005 ","End":"07:47.030","Text":"But we don\u0027t know what kind it is,"},{"Start":"07:47.030 ","End":"07:50.960","Text":"whether it\u0027s maximum, minimum or saddle."},{"Start":"07:50.960 ","End":"07:57.795","Text":"What we want to do is substitute it in the what we called"},{"Start":"07:57.795 ","End":"08:05.990","Text":"D. I brought it in the formula in general for D we\u0027ll need that."},{"Start":"08:05.990 ","End":"08:08.045","Text":"Let\u0027s compute."},{"Start":"08:08.045 ","End":"08:12.080","Text":"Well, F here takes the place of S. In our case,"},{"Start":"08:12.080 ","End":"08:15.319","Text":"we have S, S,"},{"Start":"08:15.319 ","End":"08:19.265","Text":"S. It just as a general formula with a function f,"},{"Start":"08:19.265 ","End":"08:29.680","Text":"we get S_xx at the point 4,4 is equal to just x equals 4,4 cubed is 64 this is 2."},{"Start":"08:29.680 ","End":"08:32.010","Text":"128 of 64 is 2."},{"Start":"08:32.010 ","End":"08:33.915","Text":"Similarly, y is 4,"},{"Start":"08:33.915 ","End":"08:39.160","Text":"so S_yy is also 2."},{"Start":"08:39.230 ","End":"08:41.670","Text":"This is equal to 1."},{"Start":"08:41.670 ","End":"08:48.140","Text":"I\u0027ll just copy it again as xy at the point 4,4 equals 1."},{"Start":"08:48.140 ","End":"08:54.860","Text":"I forgot to write here that this at 4,4 is equal to 2."},{"Start":"08:54.860 ","End":"08:58.110","Text":"Now if I look at this expression,"},{"Start":"08:58.110 ","End":"09:04.370","Text":"this looks a mess with the S. We just understand that S is in"},{"Start":"09:04.370 ","End":"09:10.970","Text":"the place of F. So this at our point is equal to 2."},{"Start":"09:10.970 ","End":"09:13.954","Text":"This is equal to 2,"},{"Start":"09:13.954 ","End":"09:16.040","Text":"and this is 1 squared."},{"Start":"09:16.040 ","End":"09:20.640","Text":"Basically, D at the point"},{"Start":"09:20.640 ","End":"09:27.379","Text":"4,4 is equal to 2 times 2 minus 1 squared,"},{"Start":"09:27.379 ","End":"09:30.695","Text":"which is 3, which is bigger than 0,"},{"Start":"09:30.695 ","End":"09:34.530","Text":"which means that we know it\u0027s an extremum."},{"Start":"09:34.970 ","End":"09:38.990","Text":"But to find out whether it\u0027s a maximum or minimum,"},{"Start":"09:38.990 ","End":"09:44.570","Text":"we look at the second partial derivative with respect to x."},{"Start":"09:44.570 ","End":"09:53.540","Text":"This is 2 and this is also positive and when D is positive and S_xx is positive,"},{"Start":"09:53.540 ","End":"09:57.400","Text":"that\u0027s the condition if you look it up that that\u0027s a minimum."},{"Start":"09:57.400 ","End":"10:01.800","Text":"In other words, if D is bigger than 0,"},{"Start":"10:01.800 ","End":"10:04.230","Text":"S_xx or F_xx, in general,"},{"Start":"10:04.230 ","End":"10:06.030","Text":"is bigger than 0."},{"Start":"10:06.030 ","End":"10:10.650","Text":"These 2 together give us a minimum."},{"Start":"10:13.120 ","End":"10:16.220","Text":"Let\u0027s see that we answer the question."},{"Start":"10:16.220 ","End":"10:18.320","Text":"We know that x equals 4, y equals 4."},{"Start":"10:18.320 ","End":"10:19.400","Text":"But what is the question?"},{"Start":"10:19.400 ","End":"10:20.750","Text":"So I\u0027ll go back up."},{"Start":"10:20.750 ","End":"10:24.350","Text":"It says, compute the dimensions."},{"Start":"10:24.350 ","End":"10:34.235","Text":"For the dimensions, I can say that x or if you want to call it width equals 4,"},{"Start":"10:34.235 ","End":"10:39.050","Text":"y equals length equals 4 and z equals,"},{"Start":"10:39.050 ","End":"10:41.780","Text":"well, I just substituted here."},{"Start":"10:41.780 ","End":"10:49.130","Text":"What I get is 32 over 4 times"},{"Start":"10:49.130 ","End":"10:58.245","Text":"4 and 4 times 4 is 16,16 to 32 goes twice."},{"Start":"10:58.245 ","End":"11:05.960","Text":"If you want, you can add the words width, length, height."},{"Start":"11:05.960 ","End":"11:12.590","Text":"These are the dimensions and we should have really added the unit, centimeters."},{"Start":"11:12.590 ","End":"11:15.120","Text":"Centimeters."},{"Start":"11:16.790 ","End":"11:19.990","Text":"I believe we\u0027re done."}],"ID":9025},{"Watched":false,"Name":"Exercise 11","Duration":"12m 41s","ChapterTopicVideoID":8743,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8743.jpeg","UploadDate":"2017-02-13T06:20:38.7370000","DurationForVideoObject":"PT12M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this exercise, we have to find"},{"Start":"00:03.000 ","End":"00:08.204","Text":"the shortest distance from a given point to a given plane."},{"Start":"00:08.204 ","End":"00:09.750","Text":"But not just the distance,"},{"Start":"00:09.750 ","End":"00:14.295","Text":"you want the point on the plane which that shortest distance is met."},{"Start":"00:14.295 ","End":"00:18.075","Text":"Let me introduce a diagram."},{"Start":"00:18.075 ","End":"00:20.970","Text":"This diagram illustrates the general concept."},{"Start":"00:20.970 ","End":"00:27.240","Text":"In our case, the point is going to be the point 1, 2, 3."},{"Start":"00:27.240 ","End":"00:37.650","Text":"This plane is this plane minus 2x minus 2y plus z equals 0."},{"Start":"00:37.650 ","End":"00:41.680","Text":"A is minus 2, b is, and so on."},{"Start":"00:41.750 ","End":"00:50.600","Text":"Now, if d is the distance between the point and general point x, y,"},{"Start":"00:50.600 ","End":"00:54.890","Text":"z, let\u0027s say this is not the one that\u0027s necessarily closest,"},{"Start":"00:54.890 ","End":"00:59.620","Text":"but a general point and call it x, y, z."},{"Start":"00:59.620 ","End":"01:06.040","Text":"Then the distance which we\u0027ll call d is given by,"},{"Start":"01:06.040 ","End":"01:09.640","Text":"using the extended Pythagoras theorem,"},{"Start":"01:09.640 ","End":"01:18.350","Text":"x minus 1, the x of the point squared plus y minus the y of the point,"},{"Start":"01:18.350 ","End":"01:19.999","Text":"which is 2 squared,"},{"Start":"01:19.999 ","End":"01:25.995","Text":"plus z minus the z of the point squared."},{"Start":"01:25.995 ","End":"01:28.040","Text":"Now, I\u0027m going to introduce a trick."},{"Start":"01:28.040 ","End":"01:32.165","Text":"It\u0027s a useful trick that\u0027s commonly used in mathematics,"},{"Start":"01:32.165 ","End":"01:34.040","Text":"especially with square roots."},{"Start":"01:34.040 ","End":"01:38.750","Text":"Instead of looking for the least distance and having to mess with square roots,"},{"Start":"01:38.750 ","End":"01:43.770","Text":"what if I define D as the distance squared?"},{"Start":"01:44.150 ","End":"01:47.145","Text":"Then when the distance is least,"},{"Start":"01:47.145 ","End":"01:49.115","Text":"so is the square of the distance."},{"Start":"01:49.115 ","End":"01:52.085","Text":"In fact, this works for any positive functions."},{"Start":"01:52.085 ","End":"01:56.640","Text":"Wouldn\u0027t work for negatives because minus 3"},{"Start":"01:56.640 ","End":"02:00.790","Text":"is less than 2 but when you square them it\u0027s not less than."},{"Start":"02:00.790 ","End":"02:02.410","Text":"But for positive functions,"},{"Start":"02:02.410 ","End":"02:07.690","Text":"you can always square it and get the same point at which the minimum is obtained."},{"Start":"02:07.690 ","End":"02:14.020","Text":"This is a useful trick because now big D is equal to without the square root x minus"},{"Start":"02:14.020 ","End":"02:21.290","Text":"1 squared plus y minus 2 squared plus z minus 3 squared."},{"Start":"02:21.290 ","End":"02:26.040","Text":"But I have here 3 variables and I\u0027d like a function of 2 variables."},{"Start":"02:26.040 ","End":"02:29.530","Text":"Besides, I haven\u0027t used the information that the point x,"},{"Start":"02:29.530 ","End":"02:31.570","Text":"y, z is on the plane."},{"Start":"02:31.570 ","End":"02:36.710","Text":"What we\u0027ll do is we\u0027ll write from here z in terms of x."},{"Start":"02:36.710 ","End":"02:40.325","Text":"If I just bring the 2x plus and 2y to the other side,"},{"Start":"02:40.325 ","End":"02:44.045","Text":"I\u0027ve got that z equals 2x plus 2y."},{"Start":"02:44.045 ","End":"02:48.590","Text":"Then I can substitute that in here for z. I\u0027ve got"},{"Start":"02:48.590 ","End":"02:53.735","Text":"big D and I\u0027ll emphasize that now it\u0027s only a function of x and y is equal to,"},{"Start":"02:53.735 ","End":"02:55.460","Text":"well, the first 2 terms are the same,"},{"Start":"02:55.460 ","End":"02:59.090","Text":"x minus 1 squared and y minus 2 squared."},{"Start":"02:59.090 ","End":"03:08.909","Text":"But here, z minus 3 will be 2x plus 2y minus 3 all squared."},{"Start":"03:08.909 ","End":"03:17.820","Text":"I\u0027m looking now for a minimum value of D. Now,"},{"Start":"03:17.820 ","End":"03:19.750","Text":"I just realized that D is a bad choice"},{"Start":"03:19.750 ","End":"03:21.820","Text":"of letter because it means something in this context."},{"Start":"03:21.820 ","End":"03:27.015","Text":"Let me change D to f so that we don\u0027t have a name clash."},{"Start":"03:27.015 ","End":"03:32.800","Text":"Now, we\u0027re going to use our usual techniques for extremum and critical points."},{"Start":"03:32.800 ","End":"03:35.455","Text":"We, first of all, take the first derivative."},{"Start":"03:35.455 ","End":"03:38.800","Text":"In fact, we\u0027re going to take the first and second-order derivatives f with"},{"Start":"03:38.800 ","End":"03:42.250","Text":"respect to x is here."},{"Start":"03:42.250 ","End":"03:47.230","Text":"From here I get 2 times x minus 1 and the inner derivative is 1."},{"Start":"03:47.230 ","End":"03:52.050","Text":"Here this is constants as nothing and here I"},{"Start":"03:52.050 ","End":"03:59.085","Text":"get twice 2x plus 2y minus 3,"},{"Start":"03:59.085 ","End":"04:02.950","Text":"but there is an inner derivative which is 2."},{"Start":"04:02.950 ","End":"04:08.725","Text":"Let me cross these 2 out and change this 2 to a 4, it\u0027ll be easier."},{"Start":"04:08.725 ","End":"04:11.040","Text":"Let\u0027s collect like terms."},{"Start":"04:11.040 ","End":"04:12.450","Text":"Here we get 2x."},{"Start":"04:12.450 ","End":"04:14.385","Text":"Here, we get 8x."},{"Start":"04:14.385 ","End":"04:16.950","Text":"So that makes it 10x."},{"Start":"04:16.950 ","End":"04:19.260","Text":"Let\u0027s see us collect ys."},{"Start":"04:19.260 ","End":"04:22.005","Text":"4 times 2 is 8y."},{"Start":"04:22.005 ","End":"04:25.415","Text":"Constants from here we have minus 2."},{"Start":"04:25.415 ","End":"04:29.375","Text":"From here we have minus 12."},{"Start":"04:29.375 ","End":"04:33.260","Text":"That makes that minus 14."},{"Start":"04:33.260 ","End":"04:40.114","Text":"Now, f with respect to y is equal to this times this is nothing."},{"Start":"04:40.114 ","End":"04:46.340","Text":"This is twice y minus 2 times 1 for the inner derivative"},{"Start":"04:46.340 ","End":"04:54.449","Text":"and plus twice 2x plus 2y minus 3."},{"Start":"04:54.449 ","End":"04:58.870","Text":"Also inner derivative is 2."},{"Start":"04:59.810 ","End":"05:04.510","Text":"This time if we combined like terms,"},{"Start":"05:04.520 ","End":"05:12.375","Text":"we\u0027ll get 8x plus 10y minus 16."},{"Start":"05:12.375 ","End":"05:16.470","Text":"Now, let\u0027s continue with second-order derivatives."},{"Start":"05:17.660 ","End":"05:22.700","Text":"We get that f_xx"},{"Start":"05:22.700 ","End":"05:29.575","Text":"equals differentiate this with respect to x and it just gives us 10."},{"Start":"05:29.575 ","End":"05:35.330","Text":"F_xy. This with respect to y is 8,"},{"Start":"05:35.330 ","End":"05:39.560","Text":"which I notice also the same as f_y with respect to x. I often"},{"Start":"05:39.560 ","End":"05:44.120","Text":"do that to check that see the 2 mixed second-order derivatives are the same."},{"Start":"05:44.120 ","End":"05:52.560","Text":"Then f_yy equals this with respect to y is 10."},{"Start":"05:52.790 ","End":"05:57.080","Text":"Then we have that funny quantity called D,"},{"Start":"05:57.080 ","End":"06:00.335","Text":"which is this times this minus this squared."},{"Start":"06:00.335 ","End":"06:07.035","Text":"It\u0027s 10 times 10 minus 8 squared."},{"Start":"06:07.035 ","End":"06:12.420","Text":"It\u0027s 100 minus 64, that\u0027s 36."},{"Start":"06:12.420 ","End":"06:18.080","Text":"Now, we can tell from these whether we have minimum, maximum, or saddle."},{"Start":"06:18.080 ","End":"06:21.035","Text":"First of all, D is bigger than 0."},{"Start":"06:21.035 ","End":"06:23.045","Text":"That means it\u0027s an extremum."},{"Start":"06:23.045 ","End":"06:25.250","Text":"That means minimum or maximum."},{"Start":"06:25.250 ","End":"06:28.590","Text":"Then we look at this and if this is bigger than 0,"},{"Start":"06:28.590 ","End":"06:30.660","Text":"the f_xx, then it\u0027s a minimum."},{"Start":"06:30.660 ","End":"06:35.775","Text":"We know that we have a minimum,"},{"Start":"06:35.775 ","End":"06:38.610","Text":"but we haven\u0027t found the point yet."},{"Start":"06:38.610 ","End":"06:40.535","Text":"When we find a point,"},{"Start":"06:40.535 ","End":"06:43.290","Text":"whatever it is, it\u0027s going to be minimum."},{"Start":"06:45.310 ","End":"06:50.540","Text":"That\u0027s because these came out constant regardless of x and y,"},{"Start":"06:50.540 ","End":"06:52.535","Text":"but we still have to find the point."},{"Start":"06:52.535 ","End":"06:55.865","Text":"We just know that it\u0027s going to be a minimum once we\u0027ve found it."},{"Start":"06:55.865 ","End":"07:00.485","Text":"We need to do f_x and f_y both 0."},{"Start":"07:00.485 ","End":"07:02.165","Text":"From here and here,"},{"Start":"07:02.165 ","End":"07:11.565","Text":"I get the equation 10x plus 8y minus 14 is 0."},{"Start":"07:11.565 ","End":"07:22.380","Text":"I\u0027ll just say equals 14 and 8x plus 10y equals 16."},{"Start":"07:22.380 ","End":"07:25.130","Text":"Now, all these coefficients are even,"},{"Start":"07:25.130 ","End":"07:27.455","Text":"so I divide everything by 2."},{"Start":"07:27.455 ","End":"07:32.915","Text":"Let me just put a line through them and say this is going to be 5, 4,"},{"Start":"07:32.915 ","End":"07:40.750","Text":"7, and this will be 4, 5, and 8."},{"Start":"07:40.750 ","End":"07:43.280","Text":"That\u0027s just making it smaller numbers."},{"Start":"07:43.280 ","End":"07:47.045","Text":"Now, we\u0027ll use the method of elimination where,"},{"Start":"07:47.045 ","End":"07:52.309","Text":"let\u0027s say, I want to get rid of y."},{"Start":"07:52.309 ","End":"08:00.045","Text":"So I can multiply this equation by 5 and this equation by 4 and we\u0027ll get,"},{"Start":"08:00.045 ","End":"08:07.675","Text":"I\u0027ll just note this I\u0027m multiplying by 5 everywhere and this I\u0027m multiplying by 4."},{"Start":"08:07.675 ","End":"08:11.010","Text":"One thing the ys I know will be equal."},{"Start":"08:11.010 ","End":"08:14.595","Text":"Here I\u0027ll get 20y and here I\u0027ll get 20y."},{"Start":"08:14.595 ","End":"08:23.910","Text":"Let\u0027s see 5 times 5 is 25x and 4 times 4 is 16x."},{"Start":"08:23.910 ","End":"08:25.140","Text":"On the right-hand side,"},{"Start":"08:25.140 ","End":"08:28.740","Text":"5 times 7 is 35,"},{"Start":"08:28.740 ","End":"08:34.215","Text":"and 4 times 8 is 32."},{"Start":"08:34.215 ","End":"08:38.125","Text":"Now, let\u0027s subtract the lower from the upper."},{"Start":"08:38.125 ","End":"08:39.825","Text":"I\u0027ll do a subtraction."},{"Start":"08:39.825 ","End":"08:46.110","Text":"Then we\u0027ll get 25 minus 16 is 9x."},{"Start":"08:46.110 ","End":"08:49.320","Text":"35 minus 32 is 3."},{"Start":"08:49.320 ","End":"08:56.700","Text":"9x equals 3. So x equals 1/3, 3 over 9."},{"Start":"08:56.700 ","End":"08:59.010","Text":"Now, let\u0027s find y."},{"Start":"08:59.010 ","End":"09:01.620","Text":"Choose one of these 2 to substitute in,"},{"Start":"09:01.620 ","End":"09:03.090","Text":"let\u0027s say, the top one."},{"Start":"09:03.090 ","End":"09:06.360","Text":"5x plus 4y equals 7."},{"Start":"09:06.360 ","End":"09:15.880","Text":"I get 5 times 1 third plus 4y equals 7."},{"Start":"09:16.670 ","End":"09:19.545","Text":"If I compute this,"},{"Start":"09:19.545 ","End":"09:23.205","Text":"I\u0027ll bring the 5/3 to the other side."},{"Start":"09:23.205 ","End":"09:30.355","Text":"I\u0027ve got 7 minus 5/3, y equals."},{"Start":"09:30.355 ","End":"09:33.255","Text":"Then I need to divide it by 4,"},{"Start":"09:33.255 ","End":"09:36.960","Text":"multiply top and bottom by 3,"},{"Start":"09:36.960 ","End":"09:45.420","Text":"and then I have 7 times 3 is 21 minus 5 over 12."},{"Start":"09:45.420 ","End":"09:48.180","Text":"I have 16 over 12."},{"Start":"09:48.180 ","End":"09:50.685","Text":"This comes out to 4 over 3."},{"Start":"09:50.685 ","End":"09:54.725","Text":"I\u0027ll just summarize that."},{"Start":"09:54.725 ","End":"10:01.675","Text":"So far we have x equals 1/3, y equals 4/3."},{"Start":"10:01.675 ","End":"10:06.060","Text":"Now, we still need what z equals."},{"Start":"10:06.060 ","End":"10:12.735","Text":"We substitute in here 2x plus 2y."},{"Start":"10:12.735 ","End":"10:15.525","Text":"I\u0027ll do that at the side."},{"Start":"10:15.525 ","End":"10:23.930","Text":"2 times 1/3 plus 2 times 4/3,"},{"Start":"10:23.930 ","End":"10:26.045","Text":"that\u0027s the 2x plus 2y."},{"Start":"10:26.045 ","End":"10:29.600","Text":"This comes out to be it\u0027s all over 3."},{"Start":"10:29.600 ","End":"10:30.810","Text":"2 times 1 is 2."},{"Start":"10:30.810 ","End":"10:32.880","Text":"2 times 4 is 8."},{"Start":"10:32.880 ","End":"10:38.020","Text":"2 plus 8 over 3, 10 over 3."},{"Start":"10:45.080 ","End":"10:47.525","Text":"I\u0027ll highlight it."},{"Start":"10:47.525 ","End":"10:52.650","Text":"But that\u0027s not the end of the question because if we look back up,"},{"Start":"10:53.300 ","End":"10:57.560","Text":"we\u0027ve just answered the second part actually,"},{"Start":"10:57.560 ","End":"10:59.795","Text":"the point on the plane closest,"},{"Start":"10:59.795 ","End":"11:03.240","Text":"but we haven\u0027t found the shortest distance."},{"Start":"11:03.320 ","End":"11:07.175","Text":"Now, we just need to compute the distance,"},{"Start":"11:07.175 ","End":"11:09.695","Text":"which is from here."},{"Start":"11:09.695 ","End":"11:20.340","Text":"We\u0027ve got that d or I could say d minimum is the square root."},{"Start":"11:20.500 ","End":"11:23.330","Text":"Now instead of x, y, z,"},{"Start":"11:23.330 ","End":"11:27.570","Text":"I put 1/3, 4/3, 10/3."},{"Start":"11:27.570 ","End":"11:35.615","Text":"It\u0027s 1/3 minus 1 squared plus 4/3 minus 2 squared"},{"Start":"11:35.615 ","End":"11:45.465","Text":"plus 10/3 minus 3 squared to make this longer."},{"Start":"11:45.465 ","End":"11:50.130","Text":"Let\u0027s see. 1/3 minus 1 is minus 2/3,"},{"Start":"11:50.130 ","End":"11:51.570","Text":"so when I square it,"},{"Start":"11:51.570 ","End":"12:01.720","Text":"it becomes 4/9 plus 4/3 minus 2 is minus 2/3."},{"Start":"12:01.720 ","End":"12:03.440","Text":"But again, when I square it,"},{"Start":"12:03.440 ","End":"12:06.550","Text":"it\u0027s going to be plus 4/9."},{"Start":"12:06.550 ","End":"12:09.240","Text":"10 over 3 minus 3,"},{"Start":"12:09.240 ","End":"12:12.225","Text":"it\u0027s 3 1/3 minus 3, it\u0027s 1/3 squared."},{"Start":"12:12.225 ","End":"12:16.265","Text":"It\u0027s 1/9, all this square root."},{"Start":"12:16.265 ","End":"12:19.130","Text":"But look under the root sign,"},{"Start":"12:19.130 ","End":"12:21.080","Text":"I got 4 plus 4 plus 1."},{"Start":"12:21.080 ","End":"12:23.120","Text":"9 over 9 is 1,"},{"Start":"12:23.120 ","End":"12:28.420","Text":"so it\u0027s the square root of 1, which is 1."},{"Start":"12:28.420 ","End":"12:31.160","Text":"I found the point it\u0027s x, y, and z,"},{"Start":"12:31.160 ","End":"12:37.055","Text":"and I found the shortest distance to be equal to 1."},{"Start":"12:37.055 ","End":"12:40.560","Text":"That answers all parts of the question."}],"ID":9026},{"Watched":false,"Name":"Exercise 12","Duration":"16m 40s","ChapterTopicVideoID":8744,"CourseChapterTopicPlaylistID":57492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8744.jpeg","UploadDate":"2017-02-13T06:24:34.4930000","DurationForVideoObject":"PT16M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.385","Text":"We have here an exercise."},{"Start":"00:02.385 ","End":"00:04.980","Text":"It is actually from the field of economics."},{"Start":"00:04.980 ","End":"00:09.825","Text":"But even if you\u0027re not studying economics and you\u0027re some other engineer,"},{"Start":"00:09.825 ","End":"00:13.860","Text":"that\u0027s okay, I\u0027m not getting deeply into economics."},{"Start":"00:13.860 ","End":"00:17.850","Text":"There\u0027s a manufacturer he makes calculators and he sells"},{"Start":"00:17.850 ","End":"00:21.525","Text":"them both in China and in the United States."},{"Start":"00:21.525 ","End":"00:23.625","Text":"Production costs are different."},{"Start":"00:23.625 ","End":"00:26.875","Text":"In China, it\u0027s $6 only,"},{"Start":"00:26.875 ","End":"00:31.080","Text":"and in the United States it costs him $8."},{"Start":"00:32.230 ","End":"00:38.270","Text":"He has a marketing manager and he has demands."},{"Start":"00:38.270 ","End":"00:41.270","Text":"There are demands Q_1 and Q_2,"},{"Start":"00:41.270 ","End":"00:43.970","Text":"Q stands for quantity, I guess."},{"Start":"00:43.970 ","End":"00:48.410","Text":"Quantity 1 is the number of calculators sold in"},{"Start":"00:48.410 ","End":"00:53.045","Text":"China and quantity 2 is the number of calculators sold in the United States."},{"Start":"00:53.045 ","End":"00:58.110","Text":"It\u0027s given by a formula where Q_1 is so and"},{"Start":"00:58.110 ","End":"01:02.955","Text":"so and Q_2 is so and so depending on the prices,"},{"Start":"01:02.955 ","End":"01:07.340","Text":"P for price, P_1 is the price of a calculator"},{"Start":"01:07.340 ","End":"01:14.550","Text":"in China and P_2 is in the United States."},{"Start":"01:15.580 ","End":"01:24.910","Text":"What we have to figure out is what price to charge in China and in the United States,"},{"Start":"01:24.910 ","End":"01:29.795","Text":"P_1 and P_2 in order to get the maximum profit."},{"Start":"01:29.795 ","End":"01:33.990","Text":"Furthermore, what is this maximum profit?"},{"Start":"01:33.990 ","End":"01:36.570","Text":"There\u0027s a lot of letters here,"},{"Start":"01:36.570 ","End":"01:42.410","Text":"but let\u0027s start with a very basic formula for an economics."},{"Start":"01:42.720 ","End":"01:46.510","Text":"I\u0027m going to use the letter Pi for profit,"},{"Start":"01:46.510 ","End":"01:51.700","Text":"not the Pi from geometry and the circle and all that."},{"Start":"01:51.700 ","End":"01:55.045","Text":"Just a letter Pi because P is taken for price."},{"Start":"01:55.045 ","End":"02:02.330","Text":"Price is P, profit is Pi and is going to equal total revenue."},{"Start":"02:02.330 ","End":"02:06.200","Text":"In other words, how much he takes in minus how much he spends,"},{"Start":"02:06.200 ","End":"02:08.179","Text":"which is the total cost."},{"Start":"02:08.179 ","End":"02:09.995","Text":"This is our first formula."},{"Start":"02:09.995 ","End":"02:14.600","Text":"Now, let\u0027s see if we can give what is the total revenue."},{"Start":"02:14.600 ","End":"02:19.190","Text":"The total revenue, the income is"},{"Start":"02:19.190 ","End":"02:26.240","Text":"just the revenue from China plus the revenue from the United States."},{"Start":"02:26.240 ","End":"02:34.080","Text":"In China, he has Q_1 calculators times,"},{"Start":"02:34.210 ","End":"02:40.425","Text":"I\u0027m assuming this is respectively in dollars,"},{"Start":"02:40.425 ","End":"02:45.230","Text":"P_1 dollars and in the United States,"},{"Start":"02:45.230 ","End":"02:49.040","Text":"he sells Q_2 calculators and the price of each is P_2."},{"Start":"02:49.040 ","End":"02:53.550","Text":"So this is the revenue from the United States and that\u0027s the total."},{"Start":"02:54.830 ","End":"02:59.340","Text":"Also, we have the total cost,"},{"Start":"02:59.340 ","End":"03:01.905","Text":"instead of the prices,"},{"Start":"03:01.905 ","End":"03:04.820","Text":"we\u0027re going to use the numbers 6 and 8."},{"Start":"03:04.820 ","End":"03:10.740","Text":"We have Q_1 times $6,"},{"Start":"03:10.740 ","End":"03:14.235","Text":"put the 6 first, 6 times Q_1,"},{"Start":"03:14.235 ","End":"03:20.790","Text":"because if there were Q_1 calculators in China each costing $6,"},{"Start":"03:20.790 ","End":"03:22.800","Text":"and in United States,"},{"Start":"03:22.800 ","End":"03:25.100","Text":"$8 a piece is what it costs,"},{"Start":"03:25.100 ","End":"03:26.915","Text":"and there\u0027s Q_2 of them."},{"Start":"03:26.915 ","End":"03:36.960","Text":"So altogether, we have that Pi is equal to P_1 minus 6Q_1,"},{"Start":"03:39.350 ","End":"03:43.100","Text":"I mean, I\u0027m subtracting this minus this is this."},{"Start":"03:43.100 ","End":"03:45.065","Text":"If I subtract the second,"},{"Start":"03:45.065 ","End":"03:56.100","Text":"it\u0027s P_2 minus 8Q_2."},{"Start":"03:56.100 ","End":"03:57.555","Text":"This gives me Pi,"},{"Start":"03:57.555 ","End":"04:01.800","Text":"the profit as a function of 4 variables, P_1,"},{"Start":"04:01.800 ","End":"04:06.370","Text":"Q_1, P_2, Q_2, I need a function of 2 variables."},{"Start":"04:06.370 ","End":"04:08.540","Text":"What information haven\u0027t I used?"},{"Start":"04:08.540 ","End":"04:10.565","Text":"Well, I haven\u0027t used this at all."},{"Start":"04:10.565 ","End":"04:17.030","Text":"In fact, what I can do is just substitute Q_1 from here and Q_2 from here,"},{"Start":"04:17.030 ","End":"04:19.630","Text":"I\u0027ll get everything in terms of P_1 and P_2."},{"Start":"04:19.630 ","End":"04:28.380","Text":"So what I\u0027ll get is the profit Pi as a function of the 2 sale prices."},{"Start":"04:29.390 ","End":"04:34.560","Text":"Is going to equal. Now P_1 minus 6, and instead of Q_1,"},{"Start":"04:34.560 ","End":"04:44.040","Text":"I put 116 minus 30P_1 plus 20P_2."},{"Start":"04:44.040 ","End":"04:50.550","Text":"Then the second term expands to P_2 minus 8, and instead of Q_2,"},{"Start":"04:50.550 ","End":"05:01.380","Text":"I have 144 plus 16P_1 minus 24P_2."},{"Start":"05:01.380 ","End":"05:03.050","Text":"I want to expand the brackets."},{"Start":"05:03.050 ","End":"05:09.754","Text":"Let\u0027s see if we can just do it carefully and avoid some steps."},{"Start":"05:09.754 ","End":"05:11.840","Text":"I\u0027m going to collect together."},{"Start":"05:11.840 ","End":"05:16.280","Text":"First of all, I\u0027ll go with P_1 squared terms."},{"Start":"05:16.280 ","End":"05:23.075","Text":"So here I have minus 30 and nothing from here."},{"Start":"05:23.075 ","End":"05:27.365","Text":"So I\u0027ve got minus 30P_1 squared."},{"Start":"05:27.365 ","End":"05:32.980","Text":"Let\u0027s look for terms containing P_1P_2."},{"Start":"05:32.980 ","End":"05:36.760","Text":"I\u0027ve got from here P_1 times P_2,"},{"Start":"05:36.760 ","End":"05:38.510","Text":"that\u0027s 20 of them."},{"Start":"05:38.510 ","End":"05:42.500","Text":"From here I also have P_2P_1, 16 of them,"},{"Start":"05:42.500 ","End":"05:51.275","Text":"20 from here, and 16 from here is 36P_1P_2."},{"Start":"05:51.275 ","End":"05:54.400","Text":"P_2 squared, nothing from here."},{"Start":"05:54.400 ","End":"05:58.580","Text":"P_2P_2, I\u0027ve got minus 24 of them,"},{"Start":"05:58.580 ","End":"06:02.180","Text":"minus 24 P_2 squared."},{"Start":"06:02.180 ","End":"06:04.985","Text":"These are the the quadratic terms."},{"Start":"06:04.985 ","End":"06:07.985","Text":"Now linear terms, just P_1 or just P_2."},{"Start":"06:07.985 ","End":"06:09.595","Text":"Let\u0027s start with P_1."},{"Start":"06:09.595 ","End":"06:14.745","Text":"I have P_1 times 116,"},{"Start":"06:14.745 ","End":"06:19.120","Text":"but I also have from these 2 180P_1."},{"Start":"06:20.570 ","End":"06:25.110","Text":"Let\u0027s see, how many P_1s do I have?"},{"Start":"06:25.110 ","End":"06:33.760","Text":"Again, I said 116 plus 180."},{"Start":"06:35.510 ","End":"06:42.510","Text":"From here, how many P_1 do I get just from this with this?"},{"Start":"06:42.510 ","End":"06:52.260","Text":"116 times minus 8 is minus 128."},{"Start":"06:52.260 ","End":"06:54.450","Text":"This is P_1."},{"Start":"06:54.450 ","End":"06:57.465","Text":"But let\u0027s see how many P_2 do I have?"},{"Start":"06:57.465 ","End":"07:07.200","Text":"P_2? I\u0027ve got minus 6 times 20 is minus 120."},{"Start":"07:07.200 ","End":"07:13.185","Text":"Then P_2 I can get from this times this is 144."},{"Start":"07:13.185 ","End":"07:23.370","Text":"Also from this times this 8 times 24 is 192."},{"Start":"07:23.370 ","End":"07:25.905","Text":"All this is P_2."},{"Start":"07:25.905 ","End":"07:27.455","Text":"Then just constants."},{"Start":"07:27.455 ","End":"07:32.700","Text":"Let\u0027s see, I\u0027ve got minus 6 times 116."},{"Start":"07:32.700 ","End":"07:40.105","Text":"From here, I\u0027ll put it as a minus and put them both in plus, 6 times 116,"},{"Start":"07:40.105 ","End":"07:49.890","Text":"plus 8 times 144."},{"Start":"07:50.040 ","End":"07:52.060","Text":"I did the computation."},{"Start":"07:52.060 ","End":"07:55.945","Text":"This comes out to 1,848."},{"Start":"07:55.945 ","End":"08:01.240","Text":"This comes out to 216."},{"Start":"08:01.240 ","End":"08:09.220","Text":"This comes out 168."},{"Start":"08:09.220 ","End":"08:10.720","Text":"Let\u0027s switch letters."},{"Start":"08:10.720 ","End":"08:12.790","Text":"The P_1, P_2 is quite a nuisance."},{"Start":"08:12.790 ","End":"08:15.190","Text":"Let\u0027s just call Pi,"},{"Start":"08:15.190 ","End":"08:17.050","Text":"f and we\u0027ll call P_1 and P_2,"},{"Start":"08:17.050 ","End":"08:20.200","Text":"x and y. I\u0027ll switch to a different color."},{"Start":"08:20.200 ","End":"08:22.900","Text":"We have f of x and y,"},{"Start":"08:22.900 ","End":"08:25.460","Text":"where f is just P_1,"},{"Start":"08:26.250 ","End":"08:29.605","Text":"so f is Pi and this."},{"Start":"08:29.605 ","End":"08:34.660","Text":"We get minus 30x squared plus"},{"Start":"08:34.660 ","End":"08:40.850","Text":"36xy minus 24y squared."},{"Start":"08:41.190 ","End":"08:45.430","Text":"I\u0027m missing a plus here,"},{"Start":"08:45.430 ","End":"08:55.960","Text":"plus 168x plus 216y minus 1,848."},{"Start":"08:55.960 ","End":"09:02.450","Text":"That\u0027s my function of x and y. I want to maximize this."},{"Start":"09:02.520 ","End":"09:09.025","Text":"At this point we\u0027ve moved out of economics into just the mathematics,"},{"Start":"09:09.025 ","End":"09:13.270","Text":"calculus, critical points, maximum,"},{"Start":"09:13.270 ","End":"09:14.605","Text":"minimum size, and all that."},{"Start":"09:14.605 ","End":"09:19.330","Text":"We just have to remember at the end that when we find the x and the y, and f of x, y,"},{"Start":"09:19.330 ","End":"09:24.925","Text":"we have to just re-interpreted as price 1,"},{"Start":"09:24.925 ","End":"09:26.995","Text":"price 2, and profit."},{"Start":"09:26.995 ","End":"09:30.280","Text":"Meanwhile, let\u0027s just take it as here,"},{"Start":"09:30.280 ","End":"09:32.080","Text":"we have a math problem."},{"Start":"09:32.080 ","End":"09:35.545","Text":"To maximize, we\u0027ll do it the usual way."},{"Start":"09:35.545 ","End":"09:39.850","Text":"We\u0027ll compute the partial derivatives up to second order f"},{"Start":"09:39.850 ","End":"09:44.230","Text":"with respect to x is 2 times minus 30,"},{"Start":"09:44.230 ","End":"09:46.705","Text":"so we have minus 60x."},{"Start":"09:46.705 ","End":"09:50.330","Text":"Then here plus 36y."},{"Start":"09:50.970 ","End":"09:53.140","Text":"Here nothing."},{"Start":"09:53.140 ","End":"10:02.290","Text":"Here 168, and that\u0027s it."},{"Start":"10:02.290 ","End":"10:08.900","Text":"Then f with respect to y is nothing from here."},{"Start":"10:10.920 ","End":"10:13.435","Text":"I think I miswrote this."},{"Start":"10:13.435 ","End":"10:16.310","Text":"This should be 36."},{"Start":"10:16.310 ","End":"10:19.455","Text":"Yeah, that\u0027s right. I noticed that when I came here."},{"Start":"10:19.455 ","End":"10:24.060","Text":"Here we have 36x when we differentiate with respect to y."},{"Start":"10:24.060 ","End":"10:27.840","Text":"From here, we get minus 48y,"},{"Start":"10:27.870 ","End":"10:33.325","Text":"and from here plus 216."},{"Start":"10:33.325 ","End":"10:35.320","Text":"Let\u0027s continue this first-order."},{"Start":"10:35.320 ","End":"10:40.480","Text":"Now second-order, fxx, I need fxy,"},{"Start":"10:40.480 ","End":"10:43.435","Text":"and I need fyy."},{"Start":"10:43.435 ","End":"10:50.450","Text":"Fxx, this with respect to x is just minus 60."},{"Start":"10:52.350 ","End":"10:57.490","Text":"Fxy, this with respect to y is 36,"},{"Start":"10:57.490 ","End":"11:02.590","Text":"which I noticed the same as this with respect to x. I use that as a check."},{"Start":"11:02.590 ","End":"11:07.315","Text":"Fyy is just minus 48."},{"Start":"11:07.315 ","End":"11:08.620","Text":"These are all constants."},{"Start":"11:08.620 ","End":"11:10.420","Text":"They don\u0027t depend on x and y."},{"Start":"11:10.420 ","End":"11:16.990","Text":"Actually, I can already compute the D. The D is this times this minus this squared."},{"Start":"11:16.990 ","End":"11:27.819","Text":"It\u0027s minus 60 times minus 48 minus 36 squared, and this equals,"},{"Start":"11:27.819 ","End":"11:32.680","Text":"this is going to be plus 60 times 48 is"},{"Start":"11:32.680 ","End":"11:39.205","Text":"2,880 minus 36 squared is,"},{"Start":"11:39.205 ","End":"11:43.940","Text":"if I recall, 1,296."},{"Start":"11:44.190 ","End":"11:47.590","Text":"Well, actually I don\u0027t even care what it is."},{"Start":"11:47.590 ","End":"11:49.525","Text":"It\u0027s bigger than 0."},{"Start":"11:49.525 ","End":"11:53.830","Text":"That\u0027s really all I care about if you want to actually compute it, go ahead."},{"Start":"11:53.830 ","End":"11:56.470","Text":"Now when D is bigger than 0,"},{"Start":"11:56.470 ","End":"12:00.020","Text":"then we know that we have an extremum."},{"Start":"12:00.240 ","End":"12:04.420","Text":"If we have 1 as any critical point,"},{"Start":"12:04.420 ","End":"12:06.805","Text":"we haven\u0027t found the critical points yet."},{"Start":"12:06.805 ","End":"12:08.725","Text":"I\u0027m doing things backwards."},{"Start":"12:08.725 ","End":"12:10.660","Text":"But if we do find a critical point,"},{"Start":"12:10.660 ","End":"12:12.520","Text":"we know that it\u0027s going to be an extremum."},{"Start":"12:12.520 ","End":"12:16.585","Text":"Furthermore, because fxx is negative,"},{"Start":"12:16.585 ","End":"12:19.860","Text":"we know it\u0027s going to be a maximum."},{"Start":"12:19.860 ","End":"12:24.064","Text":"It\u0027s actually going to be a maximum, not a minimum."},{"Start":"12:24.064 ","End":"12:31.460","Text":"By based on the sign of this and the table of rules of how to figure these things out."},{"Start":"12:32.550 ","End":"12:37.135","Text":"All we have to do is find the critical points which we omitted."},{"Start":"12:37.135 ","End":"12:41.785","Text":"What we do is we equate both of these to 0."},{"Start":"12:41.785 ","End":"12:43.810","Text":"I don\u0027t want to copy it again,"},{"Start":"12:43.810 ","End":"12:47.230","Text":"so I\u0027ll just put curly braces and say this equals 0,"},{"Start":"12:47.230 ","End":"12:52.540","Text":"this equals 0, 2 linear equations and 2 unknowns, x and y."},{"Start":"12:52.540 ","End":"12:55.630","Text":"Notice that all these numbers are divisible by 12."},{"Start":"12:55.630 ","End":"12:57.085","Text":"Let\u0027s reduce this."},{"Start":"12:57.085 ","End":"12:59.620","Text":"Divided by 12 is 5,"},{"Start":"12:59.620 ","End":"13:01.570","Text":"divided by 12 is 3,"},{"Start":"13:01.570 ","End":"13:08.520","Text":"divided by 12 is 14, 3, 4."},{"Start":"13:08.520 ","End":"13:13.485","Text":"See 216 divided by 12,"},{"Start":"13:13.485 ","End":"13:16.960","Text":"18. I make it."},{"Start":"13:17.480 ","End":"13:21.220","Text":"We just switch sides."},{"Start":"13:21.570 ","End":"13:26.845","Text":"I can multiply by minus and bring the numbers over to the other side."},{"Start":"13:26.845 ","End":"13:33.415","Text":"I\u0027ve got plus 5x minus 3y equals 14,"},{"Start":"13:33.415 ","End":"13:43.750","Text":"and minus 3x plus 4y minus 18 equals 0 or equals 18."},{"Start":"13:43.750 ","End":"13:46.570","Text":"That looks a lot simpler."},{"Start":"13:46.570 ","End":"13:51.445","Text":"Let\u0027s use the usual trick of equating coefficients."},{"Start":"13:51.445 ","End":"13:54.700","Text":"If I multiply, well depends what I want to get rid of."},{"Start":"13:54.700 ","End":"13:57.205","Text":"Let\u0027s say I want to get rid of y,"},{"Start":"13:57.205 ","End":"14:04.075","Text":"so I\u0027ll multiply this 1 by 4 and this 1 by 3."},{"Start":"14:04.075 ","End":"14:06.670","Text":"See what we get. If I multiply this by 4,"},{"Start":"14:06.670 ","End":"14:10.330","Text":"I\u0027ve got 20x minus 12y,"},{"Start":"14:10.330 ","End":"14:14.080","Text":"4 times 14 is 56."},{"Start":"14:14.080 ","End":"14:20.440","Text":"This by 3 minus 9x plus 12y,"},{"Start":"14:20.440 ","End":"14:23.725","Text":"18 times 3 is 54."},{"Start":"14:23.725 ","End":"14:27.520","Text":"Now what I want to do is add them because these are of opposite signs."},{"Start":"14:27.520 ","End":"14:33.085","Text":"I\u0027m going to do a plus 20 minus 9 is 11x."},{"Start":"14:33.085 ","End":"14:35.245","Text":"This plus this is nothing."},{"Start":"14:35.245 ","End":"14:39.475","Text":"This plus this gives me 110,"},{"Start":"14:39.475 ","End":"14:43.705","Text":"so x equals 10 came out a nice number."},{"Start":"14:43.705 ","End":"14:48.970","Text":"Now that I have x, I substitute it into 1 of these here."},{"Start":"14:48.970 ","End":"14:57.430","Text":"Let\u0027s take the top 1, 5x is 50 minus 3y equals 14,"},{"Start":"14:57.430 ","End":"15:04.375","Text":"so 3y is 50 minus 14,"},{"Start":"15:04.375 ","End":"15:12.115","Text":"which is 36, and so y equals 12."},{"Start":"15:12.115 ","End":"15:14.920","Text":"I\u0027ve got x, I\u0027ve got y,"},{"Start":"15:14.920 ","End":"15:20.510","Text":"and all I need now is f of x, y."},{"Start":"15:21.690 ","End":"15:26.500","Text":"Now if I substitute x and y here,"},{"Start":"15:26.500 ","End":"15:30.970","Text":"I want to figure out what is f of 10 and 12."},{"Start":"15:30.970 ","End":"15:34.345","Text":"This is just going to be a tedious computation."},{"Start":"15:34.345 ","End":"15:39.115","Text":"I will spare you the details and just give you the answer."},{"Start":"15:39.115 ","End":"15:42.955","Text":"It comes out to 288."},{"Start":"15:42.955 ","End":"15:46.550","Text":"Then finally, first of all,"},{"Start":"15:46.550 ","End":"15:51.600","Text":"I\u0027ll highlight the numbers and then we\u0027ll interpret them."},{"Start":"15:51.790 ","End":"15:54.965","Text":"Now, x was P_1,"},{"Start":"15:54.965 ","End":"16:01.655","Text":"and P_1 was the price in,"},{"Start":"16:01.655 ","End":"16:05.405","Text":"I think it was China first and US next."},{"Start":"16:05.405 ","End":"16:15.590","Text":"The price of the calculator would be $10 in China."},{"Start":"16:16.350 ","End":"16:21.530","Text":"From here I would say is price of a calculator."},{"Start":"16:21.530 ","End":"16:27.045","Text":"Write that somewhere $12 in the United States,"},{"Start":"16:27.045 ","End":"16:32.165","Text":"and the profit in doing so would be"},{"Start":"16:32.165 ","End":"16:40.530","Text":"$288 profit altogether. That\u0027s it."}],"ID":9027}],"Thumbnail":null,"ID":57492}]

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