1 00:00:00,012 --> 00:00:06,571 [music] Really, why do we have to even bother checking endpoints? 2 00:00:06,571 --> 00:00:15,235 So, let's suppose that I want to find the maximum-minimum values of this function f 3 00:00:15,235 --> 00:00:21,143 of x equals x minus x cubed. That function, f, if I considered on the 4 00:00:21,143 --> 00:00:26,097 whole real line, doesn't achieve a maximum or minimum value. 5 00:00:26,097 --> 00:00:30,077 F of x is really negative if x is really positive. 6 00:00:30,077 --> 00:00:35,423 And conversely, f of x is really positive if x is really negative. 7 00:00:35,423 --> 00:00:39,968 True enough. So, let's ask a better question. 8 00:00:39,969 --> 00:00:46,159 What if I want to maximize and minimize this function on the closed interval for 9 00:00:46,159 --> 00:00:49,975 minus 3 to 3? This is now a continuous function on a 10 00:00:49,975 --> 00:00:55,162 closed bounded interval. So, the extreme value theorem guarantees 11 00:00:55,162 --> 00:01:00,300 that I'll be successful, right? It guarantees that there is some input 12 00:01:00,300 --> 00:01:06,432 which minimizes this function's value and some input which maximizes this function's 13 00:01:06,432 --> 00:01:10,868 value on this interval. I'll begin by listing off the critical 14 00:01:10,868 --> 00:01:15,303 points for this function. So, I'll differentiate this function, 15 00:01:15,303 --> 00:01:18,555 right? And if I differentiate that function, the 16 00:01:18,555 --> 00:01:23,559 derivative of x is 1, and the derivative of minus x cubed is minus 3 x squared. 17 00:01:23,559 --> 00:01:27,951 Alright, so, there's the derivative. I'm looking for places where the 18 00:01:27,951 --> 00:01:31,827 derivative is equal to 0, or the derivative doesn't exist. 19 00:01:31,827 --> 00:01:35,151 Well, this function is differentiable everywhere. 20 00:01:35,151 --> 00:01:39,499 So, there's no critical points where the derivative doesn't exist. 21 00:01:39,499 --> 00:01:43,871 But there are going to be some places where the derivative is equal to 0. 22 00:01:43,871 --> 00:01:47,316 Let's find them now. So, I'm trying to solve this equation. 23 00:01:47,316 --> 00:01:51,669 I'll add 3x squared to both sides, I'm really trying to solve the equation, 1 24 00:01:51,669 --> 00:01:55,674 equals 3x squared. Divide both sides by 3, so now, I'm trying 25 00:01:55,674 --> 00:01:58,801 to solve the equation 1 3rd equals x squared. 26 00:01:58,801 --> 00:02:03,581 Now, I'll take a square root of both sides and I'll find that x is equal to the 27 00:02:03,581 --> 00:02:08,663 square root of a third or maybe negative the square root of a third, so I'll write 28 00:02:08,663 --> 00:02:11,574 x is plus or minus the square root of a third. 29 00:02:11,574 --> 00:02:15,012 These are the two critical points for this function. 30 00:02:15,012 --> 00:02:17,766 Oh, I'm also going to be careful to list the endpoints. 31 00:02:17,766 --> 00:02:21,037 And we'll see why we have to. So, here are the points that I should 32 00:02:21,037 --> 00:02:23,832 check, right? I should check the critical points, and I 33 00:02:23,832 --> 00:02:27,652 just found those two critical points. Negative square root of a third, and 34 00:02:27,652 --> 00:02:31,254 positive square root of a third. And I also want to check the endpoints, 35 00:02:31,254 --> 00:02:33,754 right? And my original question is asking me to 36 00:02:33,754 --> 00:02:38,234 maximize and minimize this function on this interval, which includes the endpoint 37 00:02:38,234 --> 00:02:41,627 minus 3 and the endpoint 3. So, I'm going to include those on the list 38 00:02:41,627 --> 00:02:44,888 of points to check. So, where does this continuous function 39 00:02:44,888 --> 00:02:48,333 achieve its maximum value? And where does this continuous function 40 00:02:48,333 --> 00:02:50,750 achieve its minimum value on the given domain? 41 00:02:50,750 --> 00:02:54,348 So, let's go ahead and just evaluate the function at those four points. 42 00:02:54,348 --> 00:02:58,384 Here are the endpoints, minus 3 and 3, and the critical points plus or minus the 43 00:02:58,384 --> 00:03:01,676 square root of a third. And when I evaluate the function at these 44 00:03:01,676 --> 00:03:04,814 points, what do I find? Well, I find that the largest output 45 00:03:04,814 --> 00:03:08,726 occurs when x is negative 3. And the smallest, the most negative output 46 00:03:08,726 --> 00:03:12,984 occurs when x is equal to 3. At these critical points, the outputs, you 47 00:03:12,984 --> 00:03:15,802 know, in size no bigger than 1, it's about 0.38. 48 00:03:15,802 --> 00:03:20,951 What nightmarish scenario would've ensued had I forgotten to include the endpoints? 49 00:03:20,951 --> 00:03:24,827 Well, here's the danger, right? This is actually what I'm saying. 50 00:03:24,827 --> 00:03:29,558 I'm saying that if x is in this closed interval, then the largest value of the 51 00:03:29,558 --> 00:03:34,583 function occurs when x is equal to minus 3, and the smallest value occurs when x is 52 00:03:34,583 --> 00:03:37,899 equal to 3. If I hadn't included the endpoints in my 53 00:03:37,899 --> 00:03:42,891 chart, I would have been fooled into thinking that the largest output of this 54 00:03:42,891 --> 00:03:46,931 function was 0.38. But on this interval, on this closed 55 00:03:46,931 --> 00:03:52,573 interval between minus 3 and 3, the largest output actually occurs at this 56 00:03:52,573 --> 00:03:56,211 endpoint. The upshot to all of these, is that the 57 00:03:56,211 --> 00:03:59,836 endpoints are, in some sense, critical points. 58 00:03:59,836 --> 00:04:06,232 Standing at an endpoint, you can't wiggle freely, so you really can't differentiate 59 00:04:06,232 --> 00:04:12,294 when you're standing at an endpoint. And a place where you can't differentiate 60 00:04:12,294 --> 00:04:15,000 that's one of the critical points.