1 00:00:00,012 --> 00:00:07,877 [music] Subject to some conditions functions are guaranteed to have maximum 2 00:00:07,877 --> 00:00:15,621 and minimum values here is the extreme value thery it says that if a function F 3 00:00:15,621 --> 00:00:21,127 is continuous under closed. Interval a, b, then the function attains a 4 00:00:21,127 --> 00:00:24,548 maximum value and the function attains a minimum value. 5 00:00:24,548 --> 00:00:28,446 If you don't like this, you can make it a little bit more precise. 6 00:00:28,446 --> 00:00:32,706 Here's a slightly more precise statement of the extreme value theorem. 7 00:00:32,706 --> 00:00:37,464 It says: "If a function f is continuous on the closed interval a, b," then this is 8 00:00:37,464 --> 00:00:42,144 really just making clear what I mean by, "attains a maximum value and attains a 9 00:00:42,144 --> 00:00:47,005 minimum value." All right? What I really mean is that there's some 10 00:00:47,005 --> 00:00:53,213 point, c and d In this interval so that no matter what other value I pick in that 11 00:00:53,213 --> 00:00:59,227 interval, f of c is less than or equal to that other value, and f of d is bigger 12 00:00:59,227 --> 00:01:03,777 than that other value. So f of c is the minimum value. 13 00:01:03,777 --> 00:01:10,066 C is where f achieves that minimum value. And f of d is that maximum value. 14 00:01:10,067 --> 00:01:13,900 D is where the function achieves that maximum value. 15 00:01:13,900 --> 00:01:20,098 The extreme value theorem promises us that these extreme values exist, but it doesn't 16 00:01:20,098 --> 00:01:25,642 tell us how to find those extreme values. I mean, you could have a complicated 17 00:01:25,642 --> 00:01:30,885 looking but continuous function on the closed interval between a and b. 18 00:01:30,886 --> 00:01:36,626 And it might be hard to actually determine where the maximum value occurs, but the 19 00:01:36,626 --> 00:01:42,187 Extreme Value Theorem tells you that the maximum value is achieved somewhere on 20 00:01:42,187 --> 00:01:46,843 that closed interval. Nevertheless, even in that example, there 21 00:01:46,843 --> 00:01:50,216 are extreme values, it's just hard to find them. 22 00:01:50,216 --> 00:01:55,109 Now maybe that all seems obvious. One way to understand the extreme value 23 00:01:55,109 --> 00:01:59,654 theorem is to try to break it. Try to mess with it someone and see if it 24 00:01:59,654 --> 00:02:03,049 still works. This is sort of the car method to doing 25 00:02:03,049 --> 00:02:06,704 science, right? If I'm studying cars and I want to know 26 00:02:06,704 --> 00:02:11,366 what's important to making a car go. Well, I used to just take off the wheels 27 00:02:11,366 --> 00:02:14,701 and then see if the car still works. Doesn't work so well, right? 28 00:02:14,701 --> 00:02:18,733 The fact that the car doesn't work when I break the car's wheels off tells me that 29 00:02:18,733 --> 00:02:21,041 the wheels are important for the car, right? 30 00:02:21,041 --> 00:02:24,408 Same game with this theorem. I want to figure out what the important 31 00:02:24,408 --> 00:02:28,373 parts of this theorem are so I'm going to try to break this theorem in various ways 32 00:02:28,373 --> 00:02:32,460 and if the theorem doesn't after I break it, well that must have been an important 33 00:02:32,460 --> 00:02:36,300 part of the theorem. What if, for instance, we were studying a 34 00:02:36,300 --> 00:02:39,927 function on the domain zero to infinity including zero. 35 00:02:39,927 --> 00:02:44,784 So, yeah, what if instead of closed interval between these two numbers a and 36 00:02:44,784 --> 00:02:47,773 b, I'll have infinity be one of those numbers. 37 00:02:47,773 --> 00:02:52,316 Then of course, then I have to make that an open parenthesis there. 38 00:02:52,316 --> 00:02:57,186 So if a function's just continuous on the interval between zero and infinity 39 00:02:57,186 --> 00:03:02,126 including zero, does that mean that it actually achieves its maximum minimum 40 00:03:02,126 --> 00:03:05,092 values? Well no think about this example the 41 00:03:05,092 --> 00:03:10,372 function F of X equals X define on that interval this function does not achieve a 42 00:03:10,372 --> 00:03:13,947 maximum value. If you tell me that some number is the 43 00:03:13,947 --> 00:03:19,131 maximum output of this vunction I'm just going to add one to wahtever you say and 44 00:03:19,131 --> 00:03:24,572 that'll be a bigger to this function. This function does not acheive it's maxium 45 00:03:24,572 --> 00:03:28,298 output at given input. What if we were studying a function on a 46 00:03:28,298 --> 00:03:32,146 domain that was an open interval? Yeah, so what if instead of closed 47 00:03:32,146 --> 00:03:36,189 interval, I just said open interval between these two numbers a and b? 48 00:03:36,190 --> 00:03:40,830 Well then the function would not be guaranteed to have a maximum or minimum 49 00:03:40,830 --> 00:03:44,873 value, then take a look at this. Here is an example of a continuous 50 00:03:44,873 --> 00:03:49,905 function defined at an open interval minus pi over 2 and pi over 2 and this function 51 00:03:49,905 --> 00:03:53,691 does not achieve a maximum or minimum value on this interval. 52 00:03:53,691 --> 00:03:58,285 I can make tangent as positive or as negative as I'd like by choosing x close 53 00:03:58,285 --> 00:04:02,826 enough to one of these end points. What if I try to mess up the extreme value 54 00:04:02,826 --> 00:04:07,337 theorem by eliminating the condition at the function being continuous. 55 00:04:07,337 --> 00:04:11,500 So yeah, what if I break the statement of the extreme value theorem by removing the 56 00:04:11,500 --> 00:04:15,038 continuity equation because I'm not required the function be continuous 57 00:04:15,038 --> 00:04:17,472 anymore. So, if it's just some function that's 58 00:04:17,472 --> 00:04:21,498 defined on a close interval between a and b, does it guarantee the function that 59 00:04:21,498 --> 00:04:23,929 choose the maximum and minimum value. No. 60 00:04:23,929 --> 00:04:27,071 Here's an example. Here's a picture of a graph of a 61 00:04:27,071 --> 00:04:32,391 discontinuous function and it's defined in the closed interval between a and b, but 62 00:04:32,391 --> 00:04:37,483 here you can see that it's approaching some value but never achieving that value 63 00:04:37,483 --> 00:04:42,010 and here it's approaching some value but never achieving that value. 64 00:04:42,010 --> 00:04:47,032 So this function does not Actually attain its maximum and minimum value on this 65 00:04:47,032 --> 00:04:50,211 interval. To get a better sense of what's going on, 66 00:04:50,211 --> 00:04:55,213 here's a much fancier way to talk about the possible output values of a function. 67 00:04:55,213 --> 00:04:57,954 So let's write I to represent some interval. 68 00:04:57,954 --> 00:05:02,844 It could be the open interval between a and b, the close interval between a and b, 69 00:05:02,844 --> 00:05:05,750 whatever, right? I is just a set of numbers. 70 00:05:05,751 --> 00:05:10,266 What's f of a collection of numbers? What does that even mean? 71 00:05:10,266 --> 00:05:15,411 Well this is the collection of all the output values if the input is in I. 72 00:05:15,411 --> 00:05:20,846 So f of I is the set, or collection, of all the f of xs whenever x is some point 73 00:05:20,846 --> 00:05:24,696 in the set I. And people sometimes call this the image 74 00:05:24,696 --> 00:05:27,572 of I. We can use this terminology to rephrase 75 00:05:27,572 --> 00:05:31,662 what we've been talking about. For example, the continuous image of a 76 00:05:31,662 --> 00:05:36,632 bounded interval, a bounded interval like this interval between 2 numbers, need not 77 00:05:36,632 --> 00:05:39,689 be bounded. Here's an explicit example that we've 78 00:05:39,689 --> 00:05:44,162 already seen, take a look at this continuous function, Function, tangent of 79 00:05:44,162 --> 00:05:46,712 X. Then we look at the continuous function. 80 00:05:46,712 --> 00:05:49,458 Remember to look at the image of some interval. 81 00:05:49,458 --> 00:05:53,027 Here is a interval. And what's the output of this function if 82 00:05:53,027 --> 00:05:55,957 the input is between minus pi over 2, and pi over 2. 83 00:05:55,957 --> 00:05:58,780 Well, the output can be anything. Alright? 84 00:05:58,780 --> 00:06:03,264 So f of i is all real numbers. So even though I'm starting with things 85 00:06:03,264 --> 00:06:07,143 that aren't very big. The output is as big or as small as I'd 86 00:06:07,143 --> 00:06:10,499 like it to be. What it that bounded interval is also 87 00:06:10,499 --> 00:06:13,732 closed? A much fancier way of stating the extreme 88 00:06:13,732 --> 00:06:18,716 value theorem is to state this result. That the continuous image of a closed 89 00:06:18,716 --> 00:06:21,993 bounded interval is a closed bounded interval. 90 00:06:21,993 --> 00:06:27,128 You can really try this for yourself. You write down some crazy continues 91 00:06:27,128 --> 00:06:32,879 function so who knows what this continues function might be some continues function 92 00:06:32,879 --> 00:06:37,507 and some interval which I'll call I. >> Between two numbers A and B. 93 00:06:37,507 --> 00:06:41,667 So it's bounded because A and B are numbers, right, and it's closed because 94 00:06:41,667 --> 00:06:45,539 I'm using the square brackets. And then what I want to think is what is 95 00:06:45,539 --> 00:06:47,951 all the outputs whenever the input is in I. 96 00:06:47,951 --> 00:06:50,522 Well this is going to be the closed interval. 97 00:06:50,522 --> 00:06:55,077 Right, what's the smallest value, what's the minimum value that the function 98 00:06:55,077 --> 00:06:57,936 achieves? And the biggest output is just going to be 99 00:06:57,936 --> 00:07:01,420 whatever the maximum value is that this function achieves. 100 00:07:01,420 --> 00:07:06,536 So this statement that, continuous image of a closed, bounded interval is a closed, 101 00:07:06,536 --> 00:07:09,633 bounded interval is really getting at the same idea. 102 00:07:09,633 --> 00:07:14,399 It's the extreme value theorem, saying if you start with a continuous function on 103 00:07:14,399 --> 00:07:18,736 some closed, bounded interval Then the ouptputs are going to be between. 104 00:07:18,736 --> 00:07:22,900 Including the minimum and maximum value. And the fact that these are square 105 00:07:22,900 --> 00:07:25,981 brackets here. The fact that these are closed interval 106 00:07:25,981 --> 00:07:28,947 are crucial. Because I'm telling you that the minimum 107 00:07:28,947 --> 00:07:32,423 value is actually achieved. And the maximum value is actually 108 00:07:32,423 --> 00:07:35,078 achieved. This way of talking is also useful for 109 00:07:35,078 --> 00:07:37,924 something like the intermediate value theorum. 110 00:07:37,925 --> 00:07:40,468 So, yeah. You could say it's true that the 111 00:07:40,468 --> 00:07:43,292 continuous image of an interval is an interval. 112 00:07:43,292 --> 00:07:47,783 So if you take an interval and you pick your favorite continuous function, f of 113 00:07:47,783 --> 00:07:52,742 that interval is just some other interval. That's the same as saying that over values 114 00:07:52,742 --> 00:07:57,366 in-between are achieved and that's really what the intermediate value theory is 115 00:07:57,366 --> 00:08:00,101 getting at. This is just a very succinct way of 116 00:08:00,101 --> 00:08:04,762 stating the intermediate value theorem. The upshot here is that we're seeing 117 00:08:04,762 --> 00:08:07,821 there's something really special about continuity. 118 00:08:07,821 --> 00:08:12,519 So here again is the extreme value fare I mean what we've learned is that continuity 119 00:08:12,519 --> 00:08:16,052 is really important. If someone just shows you some function 120 00:08:16,052 --> 00:08:20,132 that's not continuous don't expect that you're going to be able to find any 121 00:08:20,132 --> 00:08:24,937 extreme values they might not exist. It's also really important that this is 122 00:08:24,937 --> 00:08:30,112 about closed interval between two numbers alright a closed and bound interval and 123 00:08:30,112 --> 00:08:35,212 that's really important and if you're being asked to find maximum minimum values 124 00:08:35,212 --> 00:08:40,285 not on a closed interval say you shouldn't expect that they necessary exist. 125 00:08:40,286 --> 00:08:47,558 They might not be there. So this theorem's going to help guide you 126 00:08:47,558 --> 00:08:56,581 towards the correct answer because it tells you what sort of answer to expect.