1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,320 Your support will help MIT OpenCourseWare 4 00:00:06,320 --> 00:00:10,560 continue to offer high quality educational resources for free. 5 00:00:10,560 --> 00:00:13,300 To make a donation or view additional materials 6 00:00:13,300 --> 00:00:17,116 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,116 --> 00:00:17,740 at ocw.mit.edu. 8 00:00:28,570 --> 00:00:29,320 HERBERT GROSS: Hi. 9 00:00:29,320 --> 00:00:33,650 Today we try to wrap up our present discussion 10 00:00:33,650 --> 00:00:38,620 of partial derivatives by means of a rather practical example, 11 00:00:38,620 --> 00:00:41,110 which not only has wide application 12 00:00:41,110 --> 00:00:45,320 but which ties in many of the individual principles 13 00:00:45,320 --> 00:00:49,510 that we've talked about in the last two blocks of material. 14 00:00:49,510 --> 00:00:52,640 The particular topic that I have in mind today 15 00:00:52,640 --> 00:00:57,900 is the topic known as the theory of maxima/minima of functions 16 00:00:57,900 --> 00:00:59,770 in several variables. 17 00:00:59,770 --> 00:01:02,090 You see, in part one of our course 18 00:01:02,090 --> 00:01:03,740 we studied the special case where 19 00:01:03,740 --> 00:01:06,840 we had a function from the real numbers into the real numbers. 20 00:01:06,840 --> 00:01:09,330 And what we were looking for were 21 00:01:09,330 --> 00:01:13,420 values of the independent variable for which f 22 00:01:13,420 --> 00:01:15,920 was either maximum or minimum. 23 00:01:15,920 --> 00:01:19,600 And so a natural extension of this is simply the following: 24 00:01:19,600 --> 00:01:23,110 given a real-valued function of several real variables-- 25 00:01:23,110 --> 00:01:26,670 in other words, assume that f is a mapping from n-dimensional 26 00:01:26,670 --> 00:01:31,660 space into the real numbers, f is a function from E^n into E. 27 00:01:31,660 --> 00:01:36,560 Then the n-tuple a bar, in E sub n, is called a local maximum-- 28 00:01:36,560 --> 00:01:39,140 and I suppose we might as kill two birds with one stone 29 00:01:39,140 --> 00:01:42,370 here and put in the definition for a local minimum 30 00:01:42,370 --> 00:01:43,710 at the same time. 31 00:01:43,710 --> 00:01:46,530 It's called a local maximum of f if and only 32 00:01:46,530 --> 00:01:52,510 if there exists a neighborhood n of a bar such that f of a bar 33 00:01:52,510 --> 00:01:56,940 is greater than or equal to f of x bar for every x bar 34 00:01:56,940 --> 00:01:58,310 in the neighborhood. 35 00:01:58,310 --> 00:01:59,990 I suppose, by the way, while we're 36 00:01:59,990 --> 00:02:04,290 at this, for a local minimum the condition would be what? 37 00:02:04,290 --> 00:02:06,600 For a local minimum, instead of f of a bar 38 00:02:06,600 --> 00:02:11,020 being greater than or equal to f of x bar, it would be less than 39 00:02:11,020 --> 00:02:12,270 or equal to. 40 00:02:12,270 --> 00:02:13,700 In other words, what you're saying 41 00:02:13,700 --> 00:02:18,740 is that what you mean by a local high point or low point 42 00:02:18,740 --> 00:02:19,450 is what? 43 00:02:19,450 --> 00:02:22,610 That in a neighborhood of the point in question, 44 00:02:22,610 --> 00:02:28,260 for example, if that output exceeds 45 00:02:28,260 --> 00:02:31,130 every other possible output in a sufficiently small 46 00:02:31,130 --> 00:02:35,000 neighborhood, then we say that that particular input 47 00:02:35,000 --> 00:02:37,080 is the local maximum. 48 00:02:37,080 --> 00:02:39,530 And a similar definition for local minimum. 49 00:02:39,530 --> 00:02:41,540 In other words, again, keep in mind 50 00:02:41,540 --> 00:02:44,590 that this is precisely and almost 51 00:02:44,590 --> 00:02:47,740 a word-for-word translation of what the terms relative 52 00:02:47,740 --> 00:02:51,050 maximum and relative minimum or local max and local min 53 00:02:51,050 --> 00:02:54,210 meant for functions of a single real variable. 54 00:02:54,210 --> 00:02:58,310 Consequently, except for the fact that in several variables 55 00:02:58,310 --> 00:03:01,850 it's much more difficult to describe domains because 56 00:03:01,850 --> 00:03:03,730 of all the degrees of freedom that you have, 57 00:03:03,730 --> 00:03:07,990 one would expect that these same particular tests for membership 58 00:03:07,990 --> 00:03:10,370 would occur for functions of several variables 59 00:03:10,370 --> 00:03:13,020 when we're looking for high-low points or max-min points 60 00:03:13,020 --> 00:03:15,320 as in the case of one independent variable. 61 00:03:15,320 --> 00:03:18,150 So let's just very quickly summarize this. 62 00:03:18,150 --> 00:03:22,360 How do we test for candidates for max-min points? 63 00:03:22,360 --> 00:03:25,780 Now, the idea is simply this, that if we're 64 00:03:25,780 --> 00:03:28,840 going to have a relative high or a relative low point, 65 00:03:28,840 --> 00:03:30,730 think of the thing graphically. 66 00:03:30,730 --> 00:03:33,820 If you take a cross section with respect to any one 67 00:03:33,820 --> 00:03:35,240 of the independent variables. 68 00:03:35,240 --> 00:03:41,150 In other words, if you look at f as a function just of x_1, 69 00:03:41,150 --> 00:03:44,370 say, and hold x_2 up to x_n constant, 70 00:03:44,370 --> 00:03:47,960 then when w is viewed as a function of x_1 alone, 71 00:03:47,960 --> 00:03:51,120 then one would expect that you have what? 72 00:03:51,120 --> 00:03:56,360 That just in the w,x_1-plane, you must have a candidate, 73 00:03:56,360 --> 00:03:58,920 which means that one would expect that if you took 74 00:03:58,920 --> 00:04:02,590 the partial of f with respect to x_1, that must be 0. 75 00:04:02,590 --> 00:04:04,960 Similarly, the partial of f with respect to x_2 76 00:04:04,960 --> 00:04:06,490 must be 0, et cetera. 77 00:04:06,490 --> 00:04:09,630 The partial of f with respect to x sub n must be 0. 78 00:04:09,630 --> 00:04:13,280 In other words, to find a candidate for a max-min point, 79 00:04:13,280 --> 00:04:15,120 notice that right off the bat we're 80 00:04:15,120 --> 00:04:20,269 back to a practical application of the study of systems 81 00:04:20,269 --> 00:04:23,470 of several equations in several unknowns. 82 00:04:23,470 --> 00:04:27,000 Namely, we must solve the systems of equations 83 00:04:27,000 --> 00:04:30,470 the partial of f with respect to x_1 equals 0, et cetera, 84 00:04:30,470 --> 00:04:33,250 the partial of f with respect to x_n equals 0. 85 00:04:33,250 --> 00:04:36,460 Find simultaneous solutions, whenever 86 00:04:36,460 --> 00:04:39,160 such simultaneous solutions exist. 87 00:04:39,160 --> 00:04:41,950 Now of course, there may be points in our domain 88 00:04:41,950 --> 00:04:44,380 where the partials don't exist, just 89 00:04:44,380 --> 00:04:47,170 like there was in the case of calculus of a single variable. 90 00:04:47,170 --> 00:04:50,410 In other words, the second thing we do in testing for candidates 91 00:04:50,410 --> 00:04:55,160 is we find points in the domain where f is not differentiable. 92 00:04:55,160 --> 00:04:56,970 For example, f might not be continuous. 93 00:04:56,970 --> 00:04:58,530 f might not be defined. 94 00:04:58,530 --> 00:05:00,200 Whatever the reason is, we look to see 95 00:05:00,200 --> 00:05:02,160 where f is not differentiable. 96 00:05:02,160 --> 00:05:06,210 And all points in the domain at which f is not differentiable, 97 00:05:06,210 --> 00:05:10,030 they also become candidates for max-min points. 98 00:05:10,030 --> 00:05:13,920 And thirdly, we check the boundary of the domain of f. 99 00:05:13,920 --> 00:05:15,810 In other words, if the domain of f 100 00:05:15,810 --> 00:05:17,890 happens to be a two-dimensional region, 101 00:05:17,890 --> 00:05:20,190 we check the boundary of the region. 102 00:05:20,190 --> 00:05:23,730 Again, the reason being the same as in the calculus 103 00:05:23,730 --> 00:05:25,390 of a single real variable. 104 00:05:25,390 --> 00:05:27,730 If a function is differentiable, it 105 00:05:27,730 --> 00:05:30,290 must take on its maximum and minimum values 106 00:05:30,290 --> 00:05:32,540 some place, if the domain happens 107 00:05:32,540 --> 00:05:36,140 to be a closed set, in other words, a connected set 108 00:05:36,140 --> 00:05:37,320 with a boundary. 109 00:05:37,320 --> 00:05:39,160 And we won't go into that in any more 110 00:05:39,160 --> 00:05:41,420 detail at this particular time. 111 00:05:41,420 --> 00:05:42,900 But essentially, what do we do? 112 00:05:42,900 --> 00:05:45,670 We look at the function wherever it's differentiable, 113 00:05:45,670 --> 00:05:48,440 take all of its partials, set them equal to 0, 114 00:05:48,440 --> 00:05:52,790 solve that system simultaneously to see what values of x_1 115 00:05:52,790 --> 00:05:57,180 up to x_n give us permissible candidates for max-min points, 116 00:05:57,180 --> 00:06:01,280 we check to see where the derivative does not exist, 117 00:06:01,280 --> 00:06:03,250 and that gives us another batch of points, 118 00:06:03,250 --> 00:06:06,110 and then we check the boundary values 119 00:06:06,110 --> 00:06:08,900 to see if anything peculiar happens there. 120 00:06:08,900 --> 00:06:12,010 Same three tests as we had for calculus of a single variable. 121 00:06:12,010 --> 00:06:14,090 And why is this so closely allied 122 00:06:14,090 --> 00:06:15,700 to calculus of a single variable? 123 00:06:15,700 --> 00:06:18,560 Well, if we take the case n equals 2, 124 00:06:18,560 --> 00:06:22,000 we again get a nice geometric interpretation. 125 00:06:22,000 --> 00:06:24,570 Namely, if w is a function of x and y, 126 00:06:24,570 --> 00:06:26,700 notice again our notation for what 127 00:06:26,700 --> 00:06:28,980 happens when we have two independent variables. 128 00:06:28,980 --> 00:06:31,040 They're called x and y, and usually we 129 00:06:31,040 --> 00:06:34,740 let the dependent variable be w in that case. 130 00:06:34,740 --> 00:06:37,320 Let's take a look and see what happens over here. 131 00:06:37,320 --> 00:06:38,800 Suppose, for the sake of argument, 132 00:06:38,800 --> 00:06:42,030 that we know that we have a relative low point 133 00:06:42,030 --> 00:06:45,080 corresponding to the input a comma b. 134 00:06:45,080 --> 00:06:47,990 In other words, suppose we know that f of a comma b 135 00:06:47,990 --> 00:06:52,560 is the lowest value of f, the lowest height on the surface, 136 00:06:52,560 --> 00:06:54,840 in a sufficiently small neighborhood of the point 137 00:06:54,840 --> 00:06:55,990 a comma b. 138 00:06:55,990 --> 00:07:00,160 All I'm saying is this: take any slice whatsoever, 139 00:07:00,160 --> 00:07:05,260 any plane through the point a comma b, pick any direction s. 140 00:07:05,260 --> 00:07:07,720 Take that plane perpendicular to the xy-plane 141 00:07:07,720 --> 00:07:10,850 in the s direction, and slice the surface w 142 00:07:10,850 --> 00:07:13,260 equals f of x, y with that plane, 143 00:07:13,260 --> 00:07:16,640 and we get a slice something like this. 144 00:07:16,640 --> 00:07:19,550 Not something like this, this is the slice that we get. 145 00:07:19,550 --> 00:07:21,670 Now, let's take a look at where that low point is. 146 00:07:21,670 --> 00:07:24,750 Since that is to be a low point in the entire region, 147 00:07:24,750 --> 00:07:28,310 obviously it must be a low point, in particular, 148 00:07:28,310 --> 00:07:31,390 with respect to the particular slice that we took. 149 00:07:31,390 --> 00:07:33,420 How could it be the lowest point every place 150 00:07:33,420 --> 00:07:35,400 if it's not the lowest point with respect 151 00:07:35,400 --> 00:07:37,120 to any particular slice? 152 00:07:37,120 --> 00:07:39,580 And all we're saying then is that with respect 153 00:07:39,580 --> 00:07:43,980 to this slice, notice that w is a function of s alone. 154 00:07:43,980 --> 00:07:46,140 In other words, we can talk about the directional 155 00:07:46,140 --> 00:07:49,050 derivative df/ds. 156 00:07:49,050 --> 00:07:52,630 And again, all we're saying is that directional derivative, 157 00:07:52,630 --> 00:07:56,270 df/ds, evaluated at the point a comma b, 158 00:07:56,270 --> 00:07:59,730 must be 0 for all directions s. 159 00:07:59,730 --> 00:08:02,140 And as we have seen throughout our course, 160 00:08:02,140 --> 00:08:05,390 if f happens to be a continuously differentiable 161 00:08:05,390 --> 00:08:08,060 function of the variables x_1 up to x_n, 162 00:08:08,060 --> 00:08:12,560 then the directional derivative is determined completely 163 00:08:12,560 --> 00:08:17,220 by our knowledge of the partials with respect to x_1 up to x_n. 164 00:08:17,220 --> 00:08:21,060 And you see, as far as the theory is concerned, 165 00:08:21,060 --> 00:08:22,680 that's all there is to it. 166 00:08:22,680 --> 00:08:26,060 As the cliche goes, the rest is commentary. 167 00:08:26,060 --> 00:08:28,330 Now what kind of commentary do I mean? 168 00:08:28,330 --> 00:08:30,640 Well, among other things, once we 169 00:08:30,640 --> 00:08:34,830 have located all the particular max-min candidates-- see, 170 00:08:34,830 --> 00:08:37,140 notice why I call these things candidates. 171 00:08:37,140 --> 00:08:39,710 All we're saying is that wherever 172 00:08:39,710 --> 00:08:42,559 the system of possible derivatives, 173 00:08:42,559 --> 00:08:45,930 setting them equal to 0, yields a value, 174 00:08:45,930 --> 00:08:49,410 that point is simply a candidate for a max-min point. 175 00:08:49,410 --> 00:08:52,160 Just, again, like in the calculus of a single variable. 176 00:08:52,160 --> 00:08:56,855 If f prime of a is 0, we cannot conclude that a is a local max 177 00:08:56,855 --> 00:08:58,930 or a local min. it might be a saddle point, 178 00:08:58,930 --> 00:09:02,260 a stationary point where the thing just levels off. 179 00:09:02,260 --> 00:09:04,650 It's just that once we have the candidate, 180 00:09:04,650 --> 00:09:07,870 how do we test whether it really is a max or a min? 181 00:09:07,870 --> 00:09:12,290 Well, what we have to do is, looking at f of a comma b, 182 00:09:12,290 --> 00:09:15,240 we have to see whether that's the lowest possible value 183 00:09:15,240 --> 00:09:18,200 or the highest possible value in a sufficiently small 184 00:09:18,200 --> 00:09:19,390 neighborhood of that point. 185 00:09:19,390 --> 00:09:22,400 Well, how do you characterize a neighborhood of the point? 186 00:09:22,400 --> 00:09:25,950 You look at some nearby point, which we can denote as what? 187 00:09:25,950 --> 00:09:28,560 a plus h comma b plus k. 188 00:09:28,560 --> 00:09:31,610 As if this h and k seem strange to you, 189 00:09:31,610 --> 00:09:34,800 notice that h is often what we call delta x, 190 00:09:34,800 --> 00:09:37,330 and k is what we often call delta y. 191 00:09:37,330 --> 00:09:41,170 In other words, we look at some nearby point, a plus delta x 192 00:09:41,170 --> 00:09:43,360 comma b plus delta y. 193 00:09:43,360 --> 00:09:46,540 And what we're saying is that if this 194 00:09:46,540 --> 00:09:50,140 is to be, for example, a high point, 195 00:09:50,140 --> 00:09:52,850 it means that when you compute this difference, 196 00:09:52,850 --> 00:09:56,920 this difference had better be negative all the time. 197 00:09:56,920 --> 00:10:00,140 Because when you look at this particular thing over here, 198 00:10:00,140 --> 00:10:02,220 if this is to be the greatest possible value 199 00:10:02,220 --> 00:10:04,004 in a neighborhood when you subtract it 200 00:10:04,004 --> 00:10:05,670 from something else in the neighborhood, 201 00:10:05,670 --> 00:10:07,410 you should get a negative value. 202 00:10:07,410 --> 00:10:10,790 In other words, to put it in still other terms, what 203 00:10:10,790 --> 00:10:13,740 we're saying is that to-- let's just read this again. 204 00:10:13,740 --> 00:10:16,430 Once a max-min candidate a comma b is found, 205 00:10:16,430 --> 00:10:20,450 we must investigate the sign of f of a plus h comma b 206 00:10:20,450 --> 00:10:26,080 plus k minus f of a, b for all sufficiently small values 207 00:10:26,080 --> 00:10:28,965 of h and k. 208 00:10:28,965 --> 00:10:30,110 OK? 209 00:10:30,110 --> 00:10:32,270 That's for all sufficiently small values of h of k, 210 00:10:32,270 --> 00:10:33,150 which can be messy. 211 00:10:33,150 --> 00:10:34,040 I don't mean that. 212 00:10:34,040 --> 00:10:37,695 I mean for all sufficiently small values of h and k, 213 00:10:37,695 --> 00:10:39,070 in other words, in a neighborhood 214 00:10:39,070 --> 00:10:40,710 of the point a comma b. 215 00:10:40,710 --> 00:10:44,360 And what I'm saying is that this particular computation 216 00:10:44,360 --> 00:10:46,470 can itself be very messy. 217 00:10:46,470 --> 00:10:46,970 You see? 218 00:10:46,970 --> 00:10:50,170 This is, again, going back to this idea of how we 219 00:10:50,170 --> 00:10:51,520 invert equations and the like. 220 00:10:51,520 --> 00:10:55,260 It's difficult to compute f of a plus h comma b plus k, 221 00:10:55,260 --> 00:10:57,140 in general, if f is a messy function, 222 00:10:57,140 --> 00:10:59,664 if f is a computationally complicated thing. 223 00:10:59,664 --> 00:11:01,830 And not only that, but we may have several unknowns, 224 00:11:01,830 --> 00:11:03,310 more than two unknowns. 225 00:11:03,310 --> 00:11:06,150 And you see, this was true in one variable. 226 00:11:06,150 --> 00:11:08,610 We saw in the case of one variable that, technically 227 00:11:08,610 --> 00:11:10,930 speaking, to test whether f of a was 228 00:11:10,930 --> 00:11:13,540 a high point of a low point, we had 229 00:11:13,540 --> 00:11:17,160 to look at f of a plus h minus f of a for all sufficiently 230 00:11:17,160 --> 00:11:18,630 small values of h. 231 00:11:18,630 --> 00:11:21,810 And that could have been a messy computation too. 232 00:11:21,810 --> 00:11:24,260 Of course, the thing that happened in the single variable 233 00:11:24,260 --> 00:11:27,010 that was very helpful to us was that in the case 234 00:11:27,010 --> 00:11:29,390 of a single variable, we were often 235 00:11:29,390 --> 00:11:33,130 able to use f double prime of a as a hint. 236 00:11:33,130 --> 00:11:35,870 In other words, whenever f double prime of a 237 00:11:35,870 --> 00:11:40,250 wasn't equal to 0, we can conclude, or could conclude, 238 00:11:40,250 --> 00:11:43,580 whether f of a was a max or min, once 239 00:11:43,580 --> 00:11:45,280 we know that f prime of a was 0. 240 00:11:45,280 --> 00:11:47,430 Remember, that was that holding water 241 00:11:47,430 --> 00:11:49,100 versus spilling water routine. 242 00:11:49,100 --> 00:11:51,980 If f double prime of a was positive, 243 00:11:51,980 --> 00:11:54,190 that meant that the curve was holding water. 244 00:11:54,190 --> 00:11:57,410 Holding water meant that you had a minimum value. 245 00:11:57,410 --> 00:12:02,000 If f double prime of a was negative, 246 00:12:02,000 --> 00:12:04,160 that meant that the curve was spilling water. 247 00:12:04,160 --> 00:12:07,329 And spilling water yielded a maximum value. 248 00:12:07,329 --> 00:12:09,620 And the only problem was, is when the second derivative 249 00:12:09,620 --> 00:12:12,577 was 0, in which case the test failed. 250 00:12:12,577 --> 00:12:14,160 What did it mean that the test failed? 251 00:12:14,160 --> 00:12:16,990 When f double prime of a was 0, the only way 252 00:12:16,990 --> 00:12:20,030 we could test to see whether a was a max or a min 253 00:12:20,030 --> 00:12:21,870 on neither, meaning a saddle point, 254 00:12:21,870 --> 00:12:26,320 what was to actually look at f of a plus h minus f of a 255 00:12:26,320 --> 00:12:29,050 and see what happened in that particular case. 256 00:12:29,050 --> 00:12:33,320 Now, one would like to believe that an analogous result held 257 00:12:33,320 --> 00:12:35,520 for the case of several real variables, 258 00:12:35,520 --> 00:12:38,860 in particular for the case of two independent variables. 259 00:12:38,860 --> 00:12:41,960 The point is that in a manner of speaking, it does. 260 00:12:41,960 --> 00:12:43,980 But in another manner of speaking, 261 00:12:43,980 --> 00:12:46,300 things are much more complicated than what 262 00:12:46,300 --> 00:12:49,120 happened in the case of one independent variable. 263 00:12:49,120 --> 00:12:51,950 In particular, what goes wrong is the following, 264 00:12:51,950 --> 00:12:54,440 and that is that the second derivative, 265 00:12:54,440 --> 00:12:56,940 in the case of two independent variables, 266 00:12:56,940 --> 00:12:59,400 involves three separate partials. 267 00:12:59,400 --> 00:13:01,290 See what do you mean by a second derivative? 268 00:13:01,290 --> 00:13:03,600 You mean you must differentiate something twice. 269 00:13:03,600 --> 00:13:05,630 Well, you could've differentiated 270 00:13:05,630 --> 00:13:09,250 the function twice with respect to x at the point a comma b. 271 00:13:09,250 --> 00:13:12,280 That's what we mean by f sub xx, recall. 272 00:13:12,280 --> 00:13:15,950 Or you might have differentiated first with respect 273 00:13:15,950 --> 00:13:19,270 to x and then with respect to y, in which case 274 00:13:19,270 --> 00:13:22,870 it would have been f sub xy of a comma b. 275 00:13:22,870 --> 00:13:24,300 In fact, I should be careful here. 276 00:13:24,300 --> 00:13:25,883 There should be a fourth one here too, 277 00:13:25,883 --> 00:13:28,240 and that is f sub y,x. 278 00:13:28,240 --> 00:13:31,490 Namely, first differentiate with respect to y and then 279 00:13:31,490 --> 00:13:32,740 with respect to x. 280 00:13:32,740 --> 00:13:34,680 The reason I left that out over here 281 00:13:34,680 --> 00:13:38,780 was simply because, if we have a nice function, meaning 282 00:13:38,780 --> 00:13:41,810 one that's continuous, and the derivatives are continuous, 283 00:13:41,810 --> 00:13:45,360 and the mixed derivatives exist and are continuous, 284 00:13:45,360 --> 00:13:47,550 we showed that the order in which we 285 00:13:47,550 --> 00:13:50,740 take the partials made no difference, that f sub xy 286 00:13:50,740 --> 00:13:53,730 was equal to f sub yx. 287 00:13:53,730 --> 00:13:56,640 But getting back to this idea, the third possibility 288 00:13:56,640 --> 00:13:58,580 is that you can have differentiated twice 289 00:13:58,580 --> 00:14:03,170 with respect to y and formed and f sub yy of a comma b. 290 00:14:03,170 --> 00:14:06,150 And so the question is, with all of these different second-order 291 00:14:06,150 --> 00:14:07,770 partial derivatives floating around, 292 00:14:07,770 --> 00:14:10,560 what do you mean by the second partial derivative? 293 00:14:10,560 --> 00:14:12,880 And the key expression, and I think 294 00:14:12,880 --> 00:14:15,810 this is far from intuitive, but the key expression 295 00:14:15,810 --> 00:14:20,860 turns out to be that the determining factor is 296 00:14:20,860 --> 00:14:23,200 the second partial with respect to x, in other words f 297 00:14:23,200 --> 00:14:27,730 sub xx, multiplied by the second partial with respect to y, f 298 00:14:27,730 --> 00:14:33,200 sub yy, minus the square of the mixed partial. 299 00:14:33,200 --> 00:14:36,830 And that particular factor, the sine of that factor, 300 00:14:36,830 --> 00:14:40,830 determines whether you have a maximum point, a minimum point, 301 00:14:40,830 --> 00:14:44,270 a saddle point, or else the test might fail. 302 00:14:44,270 --> 00:14:48,840 By the way, this is a rather difficult proof to come by. 303 00:14:48,840 --> 00:14:52,040 The proof is done in chapter 18 of the Thomas text, 304 00:14:52,040 --> 00:14:54,270 and is assigned for you. 305 00:14:54,270 --> 00:14:57,300 I tried to give you learning exercises that take you 306 00:14:57,300 --> 00:14:59,320 through the proof step by step. 307 00:14:59,320 --> 00:15:03,540 And I have also included an optional supplementary lecture 308 00:15:03,540 --> 00:15:06,370 for those of you who may still have difficulty following 309 00:15:06,370 --> 00:15:08,660 both the text and the supplementary notes 310 00:15:08,660 --> 00:15:12,210 and the learning exercises, because it's all written out 311 00:15:12,210 --> 00:15:14,090 and might like to hear the thing spoken. 312 00:15:14,090 --> 00:15:17,140 What I will do is derive this for you 313 00:15:17,140 --> 00:15:19,570 in an optional lecture for those of you who want it. 314 00:15:19,570 --> 00:15:22,170 But for the time being, to give us our overview, 315 00:15:22,170 --> 00:15:24,790 let me simply state what the properties 316 00:15:24,790 --> 00:15:27,220 of this particular quantity are. 317 00:15:27,220 --> 00:15:29,900 That the main result-- and as I just write here to remind you, 318 00:15:29,900 --> 00:15:31,580 that the details are derived later, 319 00:15:31,580 --> 00:15:34,290 both in the text, in the learning exercises, 320 00:15:34,290 --> 00:15:39,220 and in an optional lecture-- that suppose we have solved 321 00:15:39,220 --> 00:15:43,220 our simultaneous system and have found 322 00:15:43,220 --> 00:15:45,670 points a comma b, where the partial 323 00:15:45,670 --> 00:15:48,570 of f with respect to x and the partial of f with respect to y 324 00:15:48,570 --> 00:15:49,820 equals 0. 325 00:15:49,820 --> 00:15:51,790 So we now have a candidate, meaning 326 00:15:51,790 --> 00:15:56,120 a comma b now is eligible to be tested to see whether it yields 327 00:15:56,120 --> 00:15:58,790 a maximum or a minimum value. 328 00:15:58,790 --> 00:16:04,190 The test turns out to be this, you compute f sub xx times 329 00:16:04,190 --> 00:16:10,330 f sub yy minus f sub xy squared at the point a comma b. 330 00:16:10,330 --> 00:16:14,680 And if that particular number turns out to be greater than 0, 331 00:16:14,680 --> 00:16:19,930 then a comma b yields a local minimum 332 00:16:19,930 --> 00:16:25,230 of f if f sub xx happens to be positive, 333 00:16:25,230 --> 00:16:30,740 and a local maximum if f sub xx happens to be negative. 334 00:16:30,740 --> 00:16:32,800 Now, the easiest way to remember that 335 00:16:32,800 --> 00:16:36,430 is to think in terms of a partial derivative, again. 336 00:16:36,430 --> 00:16:40,620 Imagine that we've sliced the surface so that we're looking 337 00:16:40,620 --> 00:16:43,440 at a cut in the wx-plane. 338 00:16:43,440 --> 00:16:49,360 In the wx-plane, notice that if the second derivative of w 339 00:16:49,360 --> 00:16:51,800 with respect to x is positive, that 340 00:16:51,800 --> 00:16:55,070 means that the curve is holding water. 341 00:16:55,070 --> 00:16:59,270 And holding water seems to indicate a minimum, you see. 342 00:16:59,270 --> 00:17:03,050 And similarly, if it's negative it's spilling water, 343 00:17:03,050 --> 00:17:05,630 and that would indicate a maximum. 344 00:17:05,630 --> 00:17:09,750 You might say, what happens if f sub xx happens to be 0. 345 00:17:09,750 --> 00:17:13,510 And the answer is, lookit, if f sub xx happens to be 0, 346 00:17:13,510 --> 00:17:16,210 this case couldn't have occurred in the first place, 347 00:17:16,210 --> 00:17:18,480 because if f sub xx happens to be 0, 348 00:17:18,480 --> 00:17:20,480 this term drops out, in which case 349 00:17:20,480 --> 00:17:25,280 this could not be a positive expression. 350 00:17:25,280 --> 00:17:28,630 With this term missing, the smallest the square can be 351 00:17:28,630 --> 00:17:37,200 is 0, and a negative of a positive or non-zero number 352 00:17:37,200 --> 00:17:38,430 can't be negative. 353 00:17:38,430 --> 00:17:40,780 In other words, you could not obey this inequality 354 00:17:40,780 --> 00:17:42,967 if f sub xx happened to be 0. 355 00:17:42,967 --> 00:17:44,550 If you want to argue, why couldn't you 356 00:17:44,550 --> 00:17:49,350 look at f sub yy instead of f sub xx, the answer is, f sub xx 357 00:17:49,350 --> 00:17:52,630 and f sub yy must both have the same sign. 358 00:17:52,630 --> 00:17:54,340 Because if we just look at this thing, 359 00:17:54,340 --> 00:17:56,880 notice that this term has to be positive. 360 00:17:56,880 --> 00:17:59,160 You're subtracting off something positive, 361 00:17:59,160 --> 00:18:01,120 therefore the term you're subtracting from 362 00:18:01,120 --> 00:18:02,340 must be positive. 363 00:18:02,340 --> 00:18:05,270 And the only way the product of two numbers can be positive 364 00:18:05,270 --> 00:18:08,370 is if each of the factors has the same sign. 365 00:18:08,370 --> 00:18:10,490 But again, I don't want to belabor that. 366 00:18:10,490 --> 00:18:13,670 I just want to go through this thing fairly rapidly with you. 367 00:18:13,670 --> 00:18:17,820 It turns out, by the way, if this key factor, this key term, 368 00:18:17,820 --> 00:18:22,610 f sub xx f sub yy minus the square of the mixed partial, 369 00:18:22,610 --> 00:18:25,000 happens to be negative, then you can be sure 370 00:18:25,000 --> 00:18:26,600 that you have a saddle point. 371 00:18:26,600 --> 00:18:28,230 In other words, what that means is 372 00:18:28,230 --> 00:18:30,390 for any neighborhood of the point 373 00:18:30,390 --> 00:18:36,080 a comma b, for some values of f, for some values of x comma 374 00:18:36,080 --> 00:18:39,290 y in that neighborhood, the function will be greater than f 375 00:18:39,290 --> 00:18:42,520 of a comma b, and for others it will be less than f 376 00:18:42,520 --> 00:18:43,390 of a comma b. 377 00:18:43,390 --> 00:18:45,640 So it's neither a max nor a min. 378 00:18:45,640 --> 00:18:48,880 And it turns out that the situation in which to test 379 00:18:48,880 --> 00:18:53,460 fails is if this particular expression happens to equal 0. 380 00:18:53,460 --> 00:18:56,530 So again, you see that from a purely mechanical point 381 00:18:56,530 --> 00:19:01,180 of view, this test is rather easy to memorize. 382 00:19:01,180 --> 00:19:03,305 The hard part is the proof. 383 00:19:03,305 --> 00:19:04,680 And that's why, as I say, there's 384 00:19:04,680 --> 00:19:07,750 extra drill on that part, if you happen to be interested in it. 385 00:19:07,750 --> 00:19:10,060 And by the way, if you're not interested in it, 386 00:19:10,060 --> 00:19:11,490 skip the proof. 387 00:19:11,490 --> 00:19:14,020 In fact, for the sake of this course, 388 00:19:14,020 --> 00:19:17,930 I am not concerned with how well you handle max-min problems. 389 00:19:17,930 --> 00:19:21,390 I'm interested more in showing you overall 390 00:19:21,390 --> 00:19:24,090 what max-min problems mean and how 391 00:19:24,090 --> 00:19:27,520 all of the principles of partial differentiation 392 00:19:27,520 --> 00:19:30,320 seem to come up in that particular application 393 00:19:30,320 --> 00:19:33,040 of max-min problems. 394 00:19:33,040 --> 00:19:34,960 So from a theoretical point of view, 395 00:19:34,960 --> 00:19:36,580 that would complete the study of how 396 00:19:36,580 --> 00:19:41,260 one handles max-min problems, except that an even more 397 00:19:41,260 --> 00:19:45,050 difficult and subtle form of computational difficulty 398 00:19:45,050 --> 00:19:49,440 comes up, in terms of some of the practical applications 399 00:19:49,440 --> 00:19:52,990 that we have, which motivate such mechanical and 400 00:19:52,990 --> 00:19:55,425 computational devices known as, for example, 401 00:19:55,425 --> 00:19:59,010 the Lagrange multipliers and things of this sort that, 402 00:19:59,010 --> 00:20:02,550 again, are discussed in the text and in the learning exercises 403 00:20:02,550 --> 00:20:04,850 and which I will not discuss in the lecture, other 404 00:20:04,850 --> 00:20:08,440 than to motivate for you why they occur. 405 00:20:08,440 --> 00:20:11,570 And let me just finish up today's lesson in terms of one 406 00:20:11,570 --> 00:20:12,620 more topic. 407 00:20:12,620 --> 00:20:14,510 And this is a bad name for this topic, 408 00:20:14,510 --> 00:20:17,800 because we've already used this word in a different context. 409 00:20:17,800 --> 00:20:20,170 But this very often happens in mathematics, 410 00:20:20,170 --> 00:20:24,020 that the same word is used in more than one context. 411 00:20:24,020 --> 00:20:26,300 But one often talks about constraints 412 00:20:26,300 --> 00:20:29,400 when one deals with functions of several variables. 413 00:20:29,400 --> 00:20:31,190 In other words, in many cases when 414 00:20:31,190 --> 00:20:33,980 one wants to maximize or minimize 415 00:20:33,980 --> 00:20:36,100 a function of several variables, it 416 00:20:36,100 --> 00:20:38,870 turns out that certain external conditions 417 00:20:38,870 --> 00:20:40,510 happen to be imposed. 418 00:20:40,510 --> 00:20:42,540 Now, if that sounds like a difficult mouthful 419 00:20:42,540 --> 00:20:45,490 to comprehend, I would like to start off 420 00:20:45,490 --> 00:20:47,540 with an example that's already occurred 421 00:20:47,540 --> 00:20:50,250 in a max-min problem in part one of our course, 422 00:20:50,250 --> 00:20:52,470 but in such a subtle way that we never 423 00:20:52,470 --> 00:20:54,960 noticed that it was really involving a function 424 00:20:54,960 --> 00:20:56,220 of several variables. 425 00:20:56,220 --> 00:20:58,291 In fact, if it hadn't have been that subtle, 426 00:20:58,291 --> 00:21:00,040 we'd have been in trouble, because in part 427 00:21:00,040 --> 00:21:02,570 one of our course, we did not talk about functions 428 00:21:02,570 --> 00:21:04,270 of more than one real variable. 429 00:21:04,270 --> 00:21:09,060 But for example, let's revisit a type of problem that says 430 00:21:09,060 --> 00:21:12,270 let's find the minimum distance, say, from the origin, 431 00:21:12,270 --> 00:21:16,230 0 comma 0, the minimum distance from the origin to the curve 432 00:21:16,230 --> 00:21:17,960 x*y equals 1. 433 00:21:17,960 --> 00:21:20,030 Well, to find that minimum distance, 434 00:21:20,030 --> 00:21:23,400 notice that what we have to do is minimize a distance 435 00:21:23,400 --> 00:21:25,970 function, namely the square of the distance-- 436 00:21:25,970 --> 00:21:28,160 I use the square simply to eliminate the square root 437 00:21:28,160 --> 00:21:31,530 sign here-- the square of the distance from the origin 438 00:21:31,530 --> 00:21:35,490 to any point x comma y is x squared plus y squared. 439 00:21:35,490 --> 00:21:37,720 Well, notice that in this form, this 440 00:21:37,720 --> 00:21:40,220 is a function of two independent variables. 441 00:21:40,220 --> 00:21:43,860 The trouble is, you don't want the distance from the origin 442 00:21:43,860 --> 00:21:45,180 to any old point. 443 00:21:45,180 --> 00:21:49,380 The point that you're investigating has to be 444 00:21:49,380 --> 00:21:51,950 on the curve x*y equals 1. 445 00:21:51,950 --> 00:21:55,410 And that means that x and y, for your investigation 446 00:21:55,410 --> 00:21:57,710 in this problem, are not independent. 447 00:21:57,710 --> 00:21:59,680 Namely, x and y are related. 448 00:21:59,680 --> 00:22:03,540 Now by the way, sometimes this equation can be so messy that 449 00:22:03,540 --> 00:22:07,000 we cannot solve for y explicitly in terms of x. 450 00:22:07,000 --> 00:22:10,200 In this particular case, as long as x is not 0, 451 00:22:10,200 --> 00:22:13,850 we can solve specifically for y in terms of x, in which case 452 00:22:13,850 --> 00:22:16,140 we get y equals 1 over x. 453 00:22:16,140 --> 00:22:17,890 And that is, in this particular case 454 00:22:17,890 --> 00:22:20,840 then, the function that we want to minimize, 455 00:22:20,840 --> 00:22:22,880 even though it looks like a function of two 456 00:22:22,880 --> 00:22:25,510 independent variables, is really a function 457 00:22:25,510 --> 00:22:28,050 of one independent variable, because since y 458 00:22:28,050 --> 00:22:32,850 is equal to 1 over x in our investigation, f of x comma y 459 00:22:32,850 --> 00:22:36,650 is really f of x comma 1 over x. 460 00:22:36,650 --> 00:22:39,040 In other words, going back to what f is explicitly 461 00:22:39,040 --> 00:22:44,570 in this case, f of x comma y is just x squared plus 1 over x 462 00:22:44,570 --> 00:22:45,970 quantity squared. 463 00:22:45,970 --> 00:22:48,970 And that in turn is also just a function of x. 464 00:22:48,970 --> 00:22:51,310 Now you see, what makes this thing more difficult 465 00:22:51,310 --> 00:22:54,300 is, first of all, we may not be dealing with a function of just 466 00:22:54,300 --> 00:22:56,300 two variables to minimize. 467 00:22:56,300 --> 00:23:01,280 And secondly, we may have a very difficult constraint imposed. 468 00:23:01,280 --> 00:23:03,190 Or what makes things even more difficult 469 00:23:03,190 --> 00:23:05,270 is that if we have to minimize a function, 470 00:23:05,270 --> 00:23:09,010 say, a five variables, there may be two or three or even 471 00:23:09,010 --> 00:23:10,780 four constraints imposed. 472 00:23:10,780 --> 00:23:13,420 And the question is, how do you maximize or minimize 473 00:23:13,420 --> 00:23:15,220 a function, taking into consideration 474 00:23:15,220 --> 00:23:18,570 the fact that there are constraints imposed? 475 00:23:18,570 --> 00:23:21,940 And this is what brings up all of the type of material 476 00:23:21,940 --> 00:23:26,110 that I gave you as exercises in the last unit, 477 00:23:26,110 --> 00:23:28,450 where we talked about the Jacobian matrix, 478 00:23:28,450 --> 00:23:31,890 and handled the inverting systems, 479 00:23:31,890 --> 00:23:35,130 and how we could solve explicitly or implicitly 480 00:23:35,130 --> 00:23:37,490 for functions, implicit function theorems, 481 00:23:37,490 --> 00:23:39,470 things that we talked about in those exercises. 482 00:23:39,470 --> 00:23:42,830 All of those things come up in solving max-min problems 483 00:23:42,830 --> 00:23:44,110 in several unknowns. 484 00:23:44,110 --> 00:23:45,990 See, more generally, what we're saying 485 00:23:45,990 --> 00:23:48,540 is we have more variables than two, 486 00:23:48,540 --> 00:23:51,380 and we have more implicit constraints, 487 00:23:51,380 --> 00:23:55,120 meaning more constraints than just a single constraint, 488 00:23:55,120 --> 00:23:59,560 and also using more variables, and also more involving 489 00:23:59,560 --> 00:24:02,630 modifying implicit that you just can't solve 490 00:24:02,630 --> 00:24:04,290 for one of the variables explicitly 491 00:24:04,290 --> 00:24:06,120 in terms of the other, even though there 492 00:24:06,120 --> 00:24:08,290 is an implicit relationship involved. 493 00:24:08,290 --> 00:24:10,880 Now, I hate to work abstractly in general, 494 00:24:10,880 --> 00:24:13,430 but in this particular lecture I'm going to do that. 495 00:24:13,430 --> 00:24:15,850 I'm going to talk with you abstractly here, 496 00:24:15,850 --> 00:24:20,480 but all of the exercises will deal with concrete situations 497 00:24:20,480 --> 00:24:22,910 so that you'll see all of the theory come alive 498 00:24:22,910 --> 00:24:23,920 in the problems. 499 00:24:23,920 --> 00:24:28,370 But because I want to give you the material as compactly 500 00:24:28,370 --> 00:24:31,370 as possible, let me just state what the situation is. 501 00:24:31,370 --> 00:24:34,680 For example, suppose we want to maximize or minimize 502 00:24:34,680 --> 00:24:37,790 a function of what appears to be three independent variables, 503 00:24:37,790 --> 00:24:39,770 say, f of x, y, z. 504 00:24:39,770 --> 00:24:41,450 And all of a sudden, somebody tells us, 505 00:24:41,450 --> 00:24:45,290 hey, in the domain that we're interested in, x, y, and z 506 00:24:45,290 --> 00:24:46,570 are not independent. 507 00:24:46,570 --> 00:24:48,150 There's a certain constraint. 508 00:24:48,150 --> 00:24:50,300 And let's write that symbolically 509 00:24:50,300 --> 00:24:53,340 as some function of x, y, and z equals 0. 510 00:24:53,340 --> 00:24:56,190 Remember, that's simply our abstract way 511 00:24:56,190 --> 00:24:59,170 of saying that there is some functional relationship that 512 00:24:59,170 --> 00:25:01,280 relates x, y, and z. 513 00:25:01,280 --> 00:25:04,440 And we'll simply call that g of x, y, z equals 0. 514 00:25:04,440 --> 00:25:07,840 Maybe I could solve for z explicitly in terms of x and y 515 00:25:07,840 --> 00:25:09,520 from this particular relationship. 516 00:25:09,520 --> 00:25:10,780 Maybe I can. 517 00:25:10,780 --> 00:25:13,230 But at any rate, what we do know from the learning 518 00:25:13,230 --> 00:25:17,340 exercises of last time, that g of x, y, z equals 0, 519 00:25:17,340 --> 00:25:21,330 will implicitly define z as some function of x and y, 520 00:25:21,330 --> 00:25:23,960 say k of x, y, except in that case 521 00:25:23,960 --> 00:25:28,060 where the partial of g with respect to z happens to be 0. 522 00:25:28,060 --> 00:25:30,110 And I put this thing in parentheses for you 523 00:25:30,110 --> 00:25:32,370 simply to give you motivation to review 524 00:25:32,370 --> 00:25:36,250 the exercises of last time if this seems a bit vague to you. 525 00:25:36,250 --> 00:25:40,050 At any rate, what this thing means in plain English 526 00:25:40,050 --> 00:25:42,750 is that subject to the constraint, 527 00:25:42,750 --> 00:25:45,830 that z is some function of x and y. 528 00:25:45,830 --> 00:25:49,350 We take this constraint, we put that back 529 00:25:49,350 --> 00:25:54,590 into our original function that we're trying to maximize. 530 00:25:54,590 --> 00:26:00,300 Notice that f of x, y, z now becomes f of x, y, k of x, y. 531 00:26:00,300 --> 00:26:02,140 See? z is k of x, y. 532 00:26:02,140 --> 00:26:04,200 If we now look at this expression, 533 00:26:04,200 --> 00:26:09,780 notice only x and y appear, so that subject to the constraint 534 00:26:09,780 --> 00:26:13,990 g of x, y, z equals 0, the function f 535 00:26:13,990 --> 00:26:16,710 is a function of only two independent variables, 536 00:26:16,710 --> 00:26:17,720 not three. 537 00:26:17,720 --> 00:26:21,992 And to indicate that, let's simply say that f of x, y, 538 00:26:21,992 --> 00:26:25,130 z-- f of x, y, k of x, y in this case-- 539 00:26:25,130 --> 00:26:27,950 is some function h of x and y. 540 00:26:27,950 --> 00:26:29,680 And what our problem is now saying 541 00:26:29,680 --> 00:26:33,190 is, minimize or maximize the function 542 00:26:33,190 --> 00:26:36,890 h, which is a function of two independent variables. 543 00:26:36,890 --> 00:26:38,720 Now notice here-- it's been quite a 544 00:26:38,720 --> 00:26:41,050 while since we've dealt with the chain rule-- 545 00:26:41,050 --> 00:26:44,180 but notice here that the chain rule now comes up 546 00:26:44,180 --> 00:26:47,960 in a very important practical application, namely, 547 00:26:47,960 --> 00:26:52,020 this looks like an eyesore f of x, y comma k of x, y. 548 00:26:52,020 --> 00:26:53,500 How can we handle that? 549 00:26:53,500 --> 00:26:55,330 Notice that another way of saying this, 550 00:26:55,330 --> 00:27:00,664 utilizing the chain rule, is to say h of x, y is f of x, y, 551 00:27:00,664 --> 00:27:05,376 z where z is some function k of x and y. 552 00:27:05,376 --> 00:27:08,160 See, f is a function of x, y and z, 553 00:27:08,160 --> 00:27:10,225 and z is a function of x and y. 554 00:27:10,225 --> 00:27:12,100 In fact, if you wanted to say it another way, 555 00:27:12,100 --> 00:27:13,310 you could say what? 556 00:27:13,310 --> 00:27:18,490 f is some function of x, y, z, where x equals x, y equals y, 557 00:27:18,490 --> 00:27:20,120 and z equals k of x, y. 558 00:27:20,120 --> 00:27:20,620 You see? 559 00:27:20,620 --> 00:27:22,620 That's your particular chain rule. 560 00:27:22,620 --> 00:27:23,820 Now, lookit. 561 00:27:23,820 --> 00:27:26,610 See, this is, again, one of the problems of mathematics, 562 00:27:26,610 --> 00:27:29,400 which I hope is crystal clear by this time. 563 00:27:29,400 --> 00:27:33,000 Granted, that we discussed the chain rule in block three. 564 00:27:33,000 --> 00:27:37,290 That's no reason why in block four we can beg off 565 00:27:37,290 --> 00:27:39,711 and say, we had a long time ago, I don't remember that. 566 00:27:39,711 --> 00:27:40,210 No. 567 00:27:40,210 --> 00:27:43,060 Hopefully, we've made the chain rule so clear 568 00:27:43,060 --> 00:27:45,640 that any time I tell you that we have to invoke it, 569 00:27:45,640 --> 00:27:48,190 you can just write it down very, very quickly. 570 00:27:48,190 --> 00:27:50,050 How does the chain rule work now? 571 00:27:50,050 --> 00:27:53,770 To differentiate this with respect 572 00:27:53,770 --> 00:27:56,060 to x, say, we take the partial of this 573 00:27:56,060 --> 00:27:59,760 with respect to x, times the partial of x with respect to x, 574 00:27:59,760 --> 00:28:01,490 plus the partial of f with respect 575 00:28:01,490 --> 00:28:03,690 to y times the partial of y with respect 576 00:28:03,690 --> 00:28:06,990 to x, plus the partial of f with respect to z times 577 00:28:06,990 --> 00:28:08,700 the partial of z with respect to x. 578 00:28:08,700 --> 00:28:11,800 In other words, the general theory hasn't changed at all. 579 00:28:11,800 --> 00:28:14,914 To maximize or minimize h, what we're going to do 580 00:28:14,914 --> 00:28:17,080 is we're going to take the partial of h with respect 581 00:28:17,080 --> 00:28:19,150 to x and set that equal to 0, we're 582 00:28:19,150 --> 00:28:21,780 going to take the partial of h with respect to y 583 00:28:21,780 --> 00:28:23,900 and set that equal to 0, and solve 584 00:28:23,900 --> 00:28:25,860 that system simultaneously. 585 00:28:25,860 --> 00:28:27,820 But the hard point computationally 586 00:28:27,820 --> 00:28:31,160 is, sure, you can say let's set the partials is equal to 0. 587 00:28:31,160 --> 00:28:33,370 But before you can do that, you had better 588 00:28:33,370 --> 00:28:35,600 be able to take the partials. 589 00:28:35,600 --> 00:28:38,170 And that's where the computationally skill comes in. 590 00:28:38,170 --> 00:28:41,780 So how do we take the partial of h with respect to x here? 591 00:28:41,780 --> 00:28:43,300 Well, we just said that. 592 00:28:43,300 --> 00:28:46,280 The partial of h with respect to x, using the chain rule, 593 00:28:46,280 --> 00:28:50,020 is f sub x times the partial of x with respect to x, plus f 594 00:28:50,020 --> 00:28:53,600 sub y times the partial of y with respect to x, plus f sub 595 00:28:53,600 --> 00:28:56,390 z times the partial of z with respect to x. 596 00:28:56,390 --> 00:29:00,460 Now notice that the partial of x with respect to x is 1. 597 00:29:00,460 --> 00:29:03,740 So that gives me f sub x over there. 598 00:29:03,740 --> 00:29:07,650 Keep in mind that whereas z is a function of x and y-- 599 00:29:07,650 --> 00:29:10,490 let's go back here and take a quick look at that-- notice 600 00:29:10,490 --> 00:29:14,140 that in our function h, we're assuming what? 601 00:29:14,140 --> 00:29:17,050 That x and y are the independent variables, 602 00:29:17,050 --> 00:29:20,100 but that z is a function of x and y. 603 00:29:20,100 --> 00:29:22,860 The point, therefore, is that since x and y 604 00:29:22,860 --> 00:29:25,680 are independent variables, by definition 605 00:29:25,680 --> 00:29:28,580 that means that the partial of y with respect to x is 0. 606 00:29:28,580 --> 00:29:31,510 In other words, the fact that y and x are independent 607 00:29:31,510 --> 00:29:35,280 means that the change in y with respect to a change in x 608 00:29:35,280 --> 00:29:38,850 is 0, because we can change x without changing y. 609 00:29:38,850 --> 00:29:42,010 And finally, given that z is k of x, y, 610 00:29:42,010 --> 00:29:45,400 we can compute the partial of z with respect to x. 611 00:29:45,400 --> 00:29:48,055 And so, what we wind up with is that the partial of h 612 00:29:48,055 --> 00:29:51,460 with respect to x is the partial of f with respect to x, 613 00:29:51,460 --> 00:29:57,020 plus the partial of f with respect to z times 614 00:29:57,020 --> 00:29:58,830 the partial of z with respect x. 615 00:29:58,830 --> 00:30:01,210 And we must set that equal to 0. 616 00:30:01,210 --> 00:30:05,240 Again, leaving the details to you as a review of the chain 617 00:30:05,240 --> 00:30:05,750 rule. 618 00:30:05,750 --> 00:30:09,090 In a similar way, we show that the partial of h with respect 619 00:30:09,090 --> 00:30:12,390 to y is the partial of f with respect to y, 620 00:30:12,390 --> 00:30:14,570 plus the partial of f with respect to z, 621 00:30:14,570 --> 00:30:16,860 times the partial of z with respect to x, 622 00:30:16,860 --> 00:30:18,670 and we set that equal to 0. 623 00:30:18,670 --> 00:30:23,110 Now, the thing to keep in mind is to observe that f of x, y, z 624 00:30:23,110 --> 00:30:24,380 was a given function. 625 00:30:24,380 --> 00:30:26,020 We know what that looks like. 626 00:30:26,020 --> 00:30:29,580 Consequently, these four quantities are known. 627 00:30:29,580 --> 00:30:33,360 The trouble is that g of x, y, z equals 0 628 00:30:33,360 --> 00:30:37,940 defines z implicitly as a function of x and y. 629 00:30:37,940 --> 00:30:39,770 And consequently, if it turns out 630 00:30:39,770 --> 00:30:43,410 that we could not have solved our system for z 631 00:30:43,410 --> 00:30:46,740 explicitly in terms of x and y, the question mark 632 00:30:46,740 --> 00:30:48,701 would be, for example-- 633 00:30:48,701 --> 00:30:49,200 I'm sorry. 634 00:30:49,200 --> 00:30:50,850 This is a misprint over here. 635 00:30:50,850 --> 00:30:53,090 This should be, of course, the partial of-- we're 636 00:30:53,090 --> 00:30:54,800 differentiate with respect to y. 637 00:30:54,800 --> 00:30:57,010 This is the partial of f with respect to y, 638 00:30:57,010 --> 00:30:59,430 plus the partial of f with respect to z, 639 00:30:59,430 --> 00:31:03,710 times the partial of z with respect to y. 640 00:31:03,710 --> 00:31:08,500 The key point over here is simply that we must know what? 641 00:31:08,500 --> 00:31:12,410 If we can't solve for z explicitly in terms of x and y, 642 00:31:12,410 --> 00:31:15,040 how do we know what these two quantities are? 643 00:31:15,040 --> 00:31:17,350 We only know them implicitly. 644 00:31:17,350 --> 00:31:19,460 And this gives us our review, again, 645 00:31:19,460 --> 00:31:22,090 of setting differentials equal to 0 and the like. 646 00:31:22,090 --> 00:31:25,610 Namely, if g of x, y, z is identically 0, 647 00:31:25,610 --> 00:31:29,580 we can equate the derivative on both sides to 0. 648 00:31:29,580 --> 00:31:31,780 And to differentiate this thing implicitly, 649 00:31:31,780 --> 00:31:34,690 it's, again, using the chain rule, we get what? 650 00:31:34,690 --> 00:31:37,720 The partial of g with respect to x, plus the partial 651 00:31:37,720 --> 00:31:40,025 of g with respect to z, times the partial of z 652 00:31:40,025 --> 00:31:42,010 with respect to x is 0. 653 00:31:42,010 --> 00:31:45,410 Notice, by the way, that g is given explicitly. 654 00:31:45,410 --> 00:31:48,600 We're told what the function g of x, y, z looks like. 655 00:31:48,600 --> 00:31:52,400 Consequently, these things here are known. 656 00:31:52,400 --> 00:31:55,630 And we can now solve for the partial of z with respect to x. 657 00:31:55,630 --> 00:31:58,570 In fact, what is the partial of z with respect to x? 658 00:31:58,570 --> 00:32:02,770 That's nothing more than what? 659 00:32:02,770 --> 00:32:05,450 It's minus the partial of g with respect 660 00:32:05,450 --> 00:32:09,740 to x divided by the partial of g with respect to z. 661 00:32:09,740 --> 00:32:12,430 And that's why the partial of g with respect to z 662 00:32:12,430 --> 00:32:15,990 had better not be 0, otherwise we wind up in trouble here. 663 00:32:15,990 --> 00:32:19,000 Similarly, we can find what the partial of z with respect to y 664 00:32:19,000 --> 00:32:19,630 is. 665 00:32:19,630 --> 00:32:21,150 That's going to turn out to be what? 666 00:32:21,150 --> 00:32:23,820 Minus the partial of g with respect 667 00:32:23,820 --> 00:32:27,860 to y divided by the partial of g with respect to z. 668 00:32:27,860 --> 00:32:31,780 Knowing what these two values are from these two equations, 669 00:32:31,780 --> 00:32:33,900 we come back into the here. 670 00:32:33,900 --> 00:32:37,430 And now we know what these are, and now 671 00:32:37,430 --> 00:32:40,550 we simply have all of the known functions 672 00:32:40,550 --> 00:32:43,890 on the right-hand side, and we solve this system. 673 00:32:43,890 --> 00:32:45,580 Now hopefully, this is one of the times 674 00:32:45,580 --> 00:32:47,390 I hope things don't sound too clear to you, 675 00:32:47,390 --> 00:32:49,570 meaning you have an overview, but you 676 00:32:49,570 --> 00:32:53,030 begin to suspect that this is a very messy situation. 677 00:32:53,030 --> 00:32:55,480 Because, you see, what I wanted you to see 678 00:32:55,480 --> 00:32:58,250 is that solving such a system like this 679 00:32:58,250 --> 00:33:00,980 can be extremely cumbersome. 680 00:33:00,980 --> 00:33:04,710 And that's why, in the exercises, we do two things. 681 00:33:04,710 --> 00:33:06,740 We have to solve systems like this 682 00:33:06,740 --> 00:33:09,160 to get the experience of seeing that there's 683 00:33:09,160 --> 00:33:12,330 a big difference between knowing theoretically 684 00:33:12,330 --> 00:33:14,880 how to solve a system of equations 685 00:33:14,880 --> 00:33:18,160 and knowing pragmatically how to carry it out. 686 00:33:18,160 --> 00:33:21,110 And secondly, we hope that some of these computations 687 00:33:21,110 --> 00:33:24,140 become cumbersome enough so that you practically 688 00:33:24,140 --> 00:33:26,530 beg to find some shortcuts. 689 00:33:26,530 --> 00:33:29,170 Because if you're not begging to find a shortcut, 690 00:33:29,170 --> 00:33:31,620 then such things as Lagrange multipliers 691 00:33:31,620 --> 00:33:33,490 and the like, which are techniques 692 00:33:33,490 --> 00:33:37,310 for solving max-min problems subject to constraints, 693 00:33:37,310 --> 00:33:40,180 that those shortcuts don't appeal to you 694 00:33:40,180 --> 00:33:41,920 and you fail to see their significance, 695 00:33:41,920 --> 00:33:44,080 and you say, why do I have to learn these things. 696 00:33:44,080 --> 00:33:46,180 Don't I have enough problems without it? 697 00:33:46,180 --> 00:33:49,300 You see, again, notice the difference between what's 698 00:33:49,300 --> 00:33:51,510 happening here abstractly and what's 699 00:33:51,510 --> 00:33:53,010 happening computationally. 700 00:33:53,010 --> 00:33:56,150 Abstractly, the theory of max-min 701 00:33:56,150 --> 00:34:00,110 is not that difficult. But computationally, to handle it, 702 00:34:00,110 --> 00:34:03,890 you have to be extremely adept at handling systems 703 00:34:03,890 --> 00:34:06,450 of n equations and n unknowns, not necessarily 704 00:34:06,450 --> 00:34:09,690 linear equations, being able to throw in constraints 705 00:34:09,690 --> 00:34:11,409 and seeing what's happening here. 706 00:34:11,409 --> 00:34:15,370 And at any rate, that's what the learning exercises 707 00:34:15,370 --> 00:34:17,929 will be all about, and the material in the text. 708 00:34:17,929 --> 00:34:20,719 And I think that's enough of a mouthful for this time. 709 00:34:20,719 --> 00:34:22,650 So until next time, good bye. 710 00:34:27,730 --> 00:34:30,090 Funding for the publication of this video 711 00:34:30,090 --> 00:34:34,969 was provided by the Gabriella and Paul Rosenbaum Foundation. 712 00:34:34,969 --> 00:34:39,150 Help OCW continue to provide free and open access to MIT 713 00:34:39,150 --> 00:34:43,070 courses by making a donation at ocw.mit.edu/donate.