1 00:00:00,012 --> 00:00:06,291 Provided some conditions are satisfied, the Extreme Value Therom guarantees the 2 00:00:06,291 --> 00:00:12,139 existence of maximum and minimum values. But how am I supposed to find those 3 00:00:12,139 --> 00:00:17,478 maximum and minimum values? We've actually already done this in some 4 00:00:17,478 --> 00:00:23,575 examples, but it's worth describing an explicit process for finding maxima and 5 00:00:23,575 --> 00:00:26,624 minima. Here's a frour-step process. 6 00:00:26,624 --> 00:00:31,303 First, differentiate your function Can find all the critical points. 7 00:00:31,303 --> 00:00:35,469 Those are places where the derivative is equal to zero or the derivative isn't 8 00:00:35,469 --> 00:00:38,128 defined. And also lists the end points if they're 9 00:00:38,128 --> 00:00:41,028 included in your domain. Check those points, right? 10 00:00:41,028 --> 00:00:45,658 Check the end points, check the critical points, and potentially you also need to 11 00:00:45,658 --> 00:00:49,575 check the limiting behavior if you're working on an open Interval. 12 00:00:49,575 --> 00:00:53,936 Let's work an example. Okay, let's work an example. 13 00:00:53,937 --> 00:01:01,219 Let's look at the function, given by the rule f of x equals 1 over, x squared minus 14 00:01:01,219 --> 00:01:03,166 1. This quantity squared. 15 00:01:03,166 --> 00:01:07,497 But let's only consider this function on a restricted domain. 16 00:01:07,497 --> 00:01:13,635 Let's consider, a maximum, minimum values of this function on the interval between 17 00:01:13,635 --> 00:01:16,025 minus 1 and 1. This open interval. 18 00:01:16,025 --> 00:01:20,616 And now I differentiate. Yeah, the first step is to differentiate 19 00:01:20,616 --> 00:01:23,922 this function. So before I differentiate it, I'm just 20 00:01:23,922 --> 00:01:26,930 going to rewrite this. Instead of 1 over x squared minus 1 21 00:01:26,930 --> 00:01:31,161 squared, I'm going to write this as x squared minus 1 to the negative 2nd power. 22 00:01:31,161 --> 00:01:35,486 It's going to make it a little bit easier for me when I think about the derivative. 23 00:01:35,486 --> 00:01:41,053 So what's the derivative of this function? By the power rule and the chain rule This 24 00:01:41,053 --> 00:01:47,455 is negative 2 times the inside, x squared minus 1 to the negative 3rd power times 25 00:01:47,455 --> 00:01:53,954 the derivative of the inside which is 2x or another way to write this is negative 2 26 00:01:53,954 --> 00:01:59,889 times 2x, that's minus 4x divided by x squared minus 1 to the third power. 27 00:01:59,889 --> 00:02:03,114 With the derivative in hand, I can find the critical points. 28 00:02:03,114 --> 00:02:07,024 Yeah, the next step is to list the critical points, and also the end points 29 00:02:07,024 --> 00:02:08,959 if they're included. So let's see. 30 00:02:08,959 --> 00:02:11,446 What are the critical points of this function? 31 00:02:11,446 --> 00:02:16,492 Will those be places where the derivative doesn't exist, or the derivative is equal 32 00:02:16,492 --> 00:02:18,557 to zero. Let's first think about when the 33 00:02:18,557 --> 00:02:21,860 derivative is equal to zero. Well, in order for that derivative to be 34 00:02:21,860 --> 00:02:24,926 equal to zero, it's a fraction. When's a fraction equal to zero? 35 00:02:24,926 --> 00:02:28,005 Well, that occurred exactly when the numerator is equal to zero. 36 00:02:28,005 --> 00:02:30,766 So I'm really just asking when is minus 4x equal to zero? 37 00:02:30,766 --> 00:02:35,484 And that's just when x is equal to 0. So the only time when the derivative is 38 00:02:35,484 --> 00:02:40,093 equal to 0 is when x is equal to 0. Now the derivative doesn't exist when x is 39 00:02:40,093 --> 00:02:44,607 equal to 1 or when x is equal to minus 1 but, you know, the functions not even 40 00:02:44,607 --> 00:02:48,737 defined in that case anyway. And I'm only considering numbers between 41 00:02:48,737 --> 00:02:51,583 minus 1 and 1 anyhow. Don't have to worry about those. 42 00:02:51,583 --> 00:02:53,863 And I don't have to worry about end points. 43 00:02:53,863 --> 00:02:57,838 Because, again, the interval that I'm considering doesn't include the end 44 00:02:57,838 --> 00:03:01,451 points, It's an open interval. So I'm just going to check x equals zero 45 00:03:01,451 --> 00:03:06,050 to figure out what's going on there. So let's think about this point where x is 46 00:03:06,050 --> 00:03:09,267 equal to zero. Well the derivative at zero is equal to 47 00:03:09,267 --> 00:03:12,023 zero. If I just plug in zero into this I can see 48 00:03:12,023 --> 00:03:14,907 that. What's the derivative at a number between 49 00:03:14,907 --> 00:03:18,365 minus 1 and zero? Well if I plug in a number between minus 1 50 00:03:18,365 --> 00:03:22,745 and zero, x squared is then between zero and 1, so x squared minus 1 is now a 51 00:03:22,745 --> 00:03:26,749 negative number cubed. The denominator is negative, but this is a 52 00:03:26,749 --> 00:03:30,154 negative number, negative 4 times a negative number. 53 00:03:30,155 --> 00:03:35,101 So we've got negative times negative divided by negative, that's negative. 54 00:03:35,101 --> 00:03:39,866 So the derivative at a number between minus 1 and 0 is negative in this case. 55 00:03:39,866 --> 00:03:44,672 And a similar kind of reasoning will show that the derivative is positive if I 56 00:03:44,672 --> 00:03:48,844 evaluate the derivative between 0 and 1. So here is the deal. 57 00:03:48,844 --> 00:03:53,484 I've got one critical point on this interval between minus 1 and 1. 58 00:03:53,484 --> 00:03:59,134 And the function's decreasing between minus 1 and 0, and it's increasing between 59 00:03:59,134 --> 00:04:02,530 0 and 1. So what does that mean about the point 0? 60 00:04:02,530 --> 00:04:08,080 Well it means that the point 0 must be a place where the function achieves a local 61 00:04:08,080 --> 00:04:11,817 minimum value. So I can summarize that here, f of 0. 62 00:04:11,818 --> 00:04:18,005 Is by, this is the first derivative test is a local minimum value. 63 00:04:18,005 --> 00:04:24,581 What about the end behavior? What happens when x is near negative 1, or 64 00:04:24,581 --> 00:04:28,602 when x is near 1? So that's really step 4 in this checklist, 65 00:04:28,602 --> 00:04:32,200 is to check the limiting or the end behavior of this function. 66 00:04:32,200 --> 00:04:35,837 What do I mean by that? I really mean look at this when X is just 67 00:04:35,837 --> 00:04:39,669 a little bit bigger than minus Y, or X is a little bit less than 1. 68 00:04:39,670 --> 00:04:42,714 Alright, that's the end behavior of this function. 69 00:04:42,714 --> 00:04:46,664 These end points are actually included so of course I can't evaluate the function 70 00:04:46,664 --> 00:04:50,407 there but I should still see what happens when X is close to these endpoints. 71 00:04:50,407 --> 00:04:53,858 Let's imagine I was trying to draw a graph of the function, alright. 72 00:04:53,858 --> 00:04:58,338 How would I know from the derivative. I know the function's decreasing here and 73 00:04:58,338 --> 00:05:01,343 increasing here. And I know the function's got a local 74 00:05:01,343 --> 00:05:04,859 minimum value here, but linking to the first derivative test. 75 00:05:04,859 --> 00:05:07,757 So the function's going down and then it's going up. 76 00:05:07,757 --> 00:05:12,488 But what happens at these two Values, when I get close to minus 1 and when I get 77 00:05:12,488 --> 00:05:15,737 close to 1. When I get close to minus 1, the function 78 00:05:15,737 --> 00:05:20,677 is very, very large, and when I get close to 1, on this side, the function's also 79 00:05:20,677 --> 00:05:24,401 very, very large. So if I imagined trying to draw a graph of 80 00:05:24,401 --> 00:05:27,975 this function? Right, it would be coming down like this, 81 00:05:27,975 --> 00:05:32,302 flattening out and then it would be getting very, very large again. 82 00:05:32,303 --> 00:05:36,780 Let's summarize the situation. This graph will help us summarize the 83 00:05:36,780 --> 00:05:39,149 situation. Alright, what do I know. 84 00:05:39,149 --> 00:05:44,314 I know that F of 0 is the smallest output, it's the minimum value that the function 85 00:05:44,314 --> 00:05:47,722 achieves on the domain, open interval minus 1 to 1. 86 00:05:47,722 --> 00:05:52,254 And there's no maximum value. By considering this limiting behavior, as 87 00:05:52,254 --> 00:05:56,455 x approached to the endpoints which aren't included in this domain. 88 00:05:56,455 --> 00:06:01,393 I can see that this function's output can be made as positive as I'd like, right. 89 00:06:01,393 --> 00:06:05,077 I can choose values of X to make the output as big as I want. 90 00:06:05,078 --> 00:06:10,265 So I can't say this function actually achieves some maximum output value. 91 00:06:10,265 --> 00:06:13,468 It doen'st achieve some maximum output value. 92 00:06:13,468 --> 00:06:17,208 But it does achieve a minimum output value right here. 93 00:06:17,208 --> 00:06:20,074 When x is equal to 0. And that output is 1. 94 00:06:20,074 --> 00:06:24,280 It's important to emphasize that the domain truly matters. 95 00:06:24,280 --> 00:06:30,307 Yeah, what if I ask to find the maximum and minimum values of the same function f 96 00:06:30,307 --> 00:06:36,404 of x is equal to one over x squared minus one squared, but a different domain?What 97 00:06:36,404 --> 00:06:41,591 if I looked on the interval from one to infinity this open interval? 98 00:06:41,591 --> 00:06:46,909 Again we start by differentiating Well of course, the derivative's the same. 99 00:06:46,909 --> 00:06:51,651 The derivative of this function is still minus 4 x over x squared minus 1 to the 100 00:06:51,651 --> 00:06:54,749 third power. Now, where are the critical points? 101 00:06:54,749 --> 00:06:59,236 The only place that this derivative is equal to 0, is when x is equal to 0. 102 00:06:59,236 --> 00:07:03,987 And the only place where the derivative doesn't exist is just places where the 103 00:07:03,987 --> 00:07:07,723 function isn't even defined, right? So on this interval. 104 00:07:07,723 --> 00:07:13,030 From one to infinity, there's no place where the derivative isn't defined or is 105 00:07:13,030 --> 00:07:16,093 equal to zero. So there's no critical points. 106 00:07:16,093 --> 00:07:20,335 So what about the behavior near the boundaries of this interval? 107 00:07:20,335 --> 00:07:25,381 What happens if X is just a little bit more than one or when X is really, really 108 00:07:25,381 --> 00:07:27,978 big? So, we working on the interval from 1 to 109 00:07:27,978 --> 00:07:29,604 infinity. This open interval. 110 00:07:29,604 --> 00:07:33,640 So, that means the limits that you'd consider are the limits, the function as x 111 00:07:33,640 --> 00:07:37,856 approaches infinity and the limit of the function as x approaches 1 from the right 112 00:07:37,856 --> 00:07:40,421 hand side. And, doing calculation I find that the 113 00:07:40,421 --> 00:07:44,079 limit of the function as x approaches infinity is 0 because, in that case the 114 00:07:44,079 --> 00:07:48,047 denominator can be made as large as I like, which makes this ratio as close to 0 115 00:07:48,047 --> 00:07:50,451 as I like. And, the limit of the function as x 116 00:07:50,451 --> 00:07:53,252 approaches 1 from the right is equal to infinity. 117 00:07:53,252 --> 00:07:58,187 Because in that case if x approaches 1 from the right this denominator can be 118 00:07:58,187 --> 00:08:02,837 made as close to 0 and positive as I'd like which makes this ratio as large as 119 00:08:02,837 --> 00:08:05,734 I'd like. So in this case this function on this 120 00:08:05,734 --> 00:08:10,174 domain doesn't take a largest value and doesn't take a smallest value. 121 00:08:10,174 --> 00:08:14,968 The function's output can be made as big as I'd like if I'm willing to choose x 122 00:08:14,968 --> 00:08:19,908 just a little bit bigger than 1 and the function's output can be made as close to 123 00:08:19,908 --> 00:08:23,441 0 as I'd like if I'm willing to choose X to be big enough. 124 00:08:23,441 --> 00:08:28,267 In other words, the domain matters. I mean this function, if I'm considering 125 00:08:28,267 --> 00:08:33,217 the funciton on this domain, from -1 to 1. It does have a smallest output value. 126 00:08:33,217 --> 00:08:37,042 And it should use its local minimum value when x is equal to 0. 127 00:08:37,042 --> 00:08:42,142 But if consider the same function. But on a different domain there's no 128 00:08:42,142 --> 00:08:48,220 critical point on this domain, right? And I studied the limiting behavior as x 129 00:08:48,220 --> 00:08:53,986 approached one, and as x approached infinity, and I found that I could make 130 00:08:53,986 --> 00:09:00,140 the output of this function as large as I'd like and as close to zero as I'd like. 131 00:09:00,140 --> 00:09:05,940 So this function doesn't achieve a maximum minimum value on this interval.