WEBVTT 00:00:00.369 --> 00:00:04.191 What I have here in yellow is the graph of y=f（x）. 00:00:04.191 --> 00:00:07.968 That here in this move color I’ve graphed y’s equal 00:00:07.968 --> 00:00:11.510 to the derivative of f, is f′(x).And then here 00:00:11.510 --> 00:00:15.839 in blue I graphed y is equal to the second derivative of our function. 00:00:15.839 --> 00:00:18.507 So this is the derivative of this, of the first 00:00:18.507 --> 00:00:23.195 derivative right over there. And we’ve already seen examples of how 00:00:23.195 --> 00:00:27.373 can we identify minimum and maximum points. Obviously, if we have a graph in front of us, 00:00:27.373 --> 00:00:30.989 it’s not hard for human brain to identify this as a local 00:00:30.989 --> 00:00:34.703 maximum point. The function might take on higher values later on. And 00:00:34.703 --> 00:00:38.593 to identify this as a local minimum point. The function might take on 00:00:38.593 --> 00:00:43.009 lower values later on. But we saw, even if we don’t have a graph in front of 00:00:43.009 --> 00:00:46.004 us, if we are able to take the derivative of the function, we might… 00:00:46.004 --> 00:00:49.702 or if we are not able to take the derivative of the function. We might be able to identify 00:00:49.702 --> 00:00:54.110 these points as maximum or minimum. The way that we did it. Ok… what are the critical 00:00:54.110 --> 00:00:57.510 points for this function. Well, critical points over the function where the function’s 00:00:57.510 --> 00:01:01.702 derivative is either undefined or zero. This the function’s derivative. 00:01:01.702 --> 00:01:06.251 It’s zero here and here. So we would call those critical points. 00:01:06.251 --> 00:01:09.501 I don’t see any undefined. Any point was the derivative’s undefined 00:01:09.501 --> 00:01:13.377 just yet. So we would call here and 00:01:13.377 --> 00:01:17.004 here, critical points. So these are candidate 00:01:17.004 --> 00:01:20.919 minimum…these are candidate points which are function might take on a minimum or 00:01:20.919 --> 00:01:24.770 maximum value. And the way that we figured out whether it was a minimum or maximum 00:01:24.770 --> 00:01:29.191 value is to look at the behavior of the derivative around that point 00:01:29.191 --> 00:01:32.442 and over here we saw the derivative is de...or the 00:01:32.442 --> 00:01:36.245 derivative is positive.The derivative is positive 00:01:36.245 --> 00:01:38.926 as we approach that point 00:01:38.926 --> 00:01:44.104 and then it becomes negative. It goes from being positive 00:01:44.104 --> 00:01:47.775 to negative as we cross that point which means that the function] 00:01:47.775 --> 00:01:52.169 was increasing. If the derivative is positive that means the function was increasing 00:01:52.169 --> 00:01:56.109 as we approach that point and then decreasing as we leave that point. 00:01:56.109 --> 00:01:59.340 Which is a pretty good way to think about this… Being a maximum point, 00:01:59.340 --> 00:02:03.503 for increasing as we approach and decreasing as we leave it. Then this is definitely going 00:02:03.503 --> 00:02:07.235 to be a maximum point. Similarly, 00:02:07.235 --> 00:02:11.772 right over here, we see that the function is negative or the derivative 00:02:11.772 --> 00:02:15.337 is negative as we approach the point which means that the 00:02:15.337 --> 00:02:18.440 function is decreasing. And we see the derivative is 00:02:18.440 --> 00:02:23.398 positive as we exit that point. We go for having a negative derivative to a positive 00:02:23.398 --> 00:02:26.711 derivative which means the function goes from decreasing to 00:02:26.711 --> 00:02:30.504 increasing right around that point, which is a pretty good indication. 00:02:30.504 --> 00:02:34.437 Or that is an indication, that this critical point is a point at which the 00:02:34.437 --> 00:02:38.169 function takes on a minimum…a minimum value. 00:02:38.169 --> 00:02:41.710 What I want do now is to extend things by using 00:02:41.710 --> 00:02:46.653 the ideal of concavity… con-ca[ei]-vity. 00:02:46.653 --> 00:02:49.774 And I know I’m mispronouncing it, maybe it’s conca[æ]vity, 00:02:49.774 --> 00:02:53.442 but new thinking about concavity. Start to look at the second 00:02:53.442 --> 00:02:57.406 derivative, it rather than kind of seeing just as transition. To think about 00:02:57.406 --> 00:03:01.399 whether this is a minimum or maximum point. So 00:03:01.399 --> 00:03:05.316 let’s think about what’s happening in this first region. This kind of …this part of 00:03:05.316 --> 00:03:09.109 the curve up here where is it looks like an arc where it’s 00:03:09.109 --> 00:03:12.773 opening downward. Where it looks kinda like an “A” without the crossbeam or upside 00:03:12.773 --> 00:03:16.902 down “U” and then we’ll think about what’s happening in this kind of upward 00:03:16.902 --> 00:03:20.842 opening “U”, part of the curve. So over this first 00:03:20.842 --> 00:03:24.903 interval right over here, if we start we get this slope is very…is 00:03:24.903 --> 00:03:28.404 very ( actually I’ll do it in the same color, exactly the same color that 00:03:28.404 --> 00:03:31.504 I used for the actual derivative) the slope is very positive 00:03:31.504 --> 00:03:35.710 ..slope is very positive. Then it becomes less positive...becomes 00:03:35.710 --> 00:03:39.650 less positive…then it becomes even less positive…becomes even less 00:03:39.650 --> 00:03:43.502 positive…and eventually gets to zero…eventually gets to zero. Then it keeps 00:03:43.502 --> 00:03:47.321 decreasing. Now becomes slightly negative…slightly negative. Then 00:03:47.321 --> 00:03:51.238 it becomes even more negative…becomes even more negative…and 00:03:51.238 --> 00:03:55.591 then it stops decreasing right around. It looks like it stops decreasing right 00:03:55.591 --> 00:03:59.807 around there. So the slope stops decreasing right around there. You see that in the red , 00:03:59.807 --> 00:04:03.148 the slope is decreasing…decreasing…decreasing……until that point and 00:04:03.148 --> 00:04:06.399 then it starts to increase. So this entire section, 00:04:06.399 --> 00:04:09.000 this entire section right over here… 00:04:09.000 --> 00:04:14.643 the slope is decreasing. “Slope… 00:04:14.643 --> 00:04:19.057 slope is decreasing” and 00:04:19.057 --> 00:04:21.850 you see it right over here when we take the derivative, the deri…ative 00:04:21.850 --> 00:04:27.147 right over here… the entire, over this entire interval is decreasing. 00:04:27.147 --> 00:04:30.729 And we also see that when we take the second derivative. If the derivative is 00:04:30.729 --> 00:04:34.569 decreasing that means that the second, the derivative of the derivative is 00:04:34.569 --> 00:04:38.169 negative and we see that is indeed the case 00:04:38.169 --> 00:04:42.477 over this entire interval. The second derivative, the second 00:04:42.477 --> 00:04:46.307 derivative is indeed negative. Now what 00:04:46.307 --> 00:04:50.110 happens as we start to transition to this upward opening ”U” part of the curve. 00:04:50.110 --> 00:04:54.307 Well here the derivative is reasonably negative, 00:04:54.307 --> 00:04:58.050 it’s reasonably negative right there. But then it starts gets…it’s 00:04:58.050 --> 00:05:00.902 still negative but it becomes less negative and less negative 00:05:00.902 --> 00:05:06.314 …then it becomes zero, 00:05:06.314 --> 00:05:09.473 it becomes zero right over here. And then it becomes more and 00:05:09.473 --> 00:05:13.903 more and more positive, and you see that right over here. So over this 00:05:13.903 --> 00:05:18.169 entire interval, the slope or the derivative is increasing. 00:05:18.169 --> 00:05:21.901 So the slope...slope is 00:05:21.901 --> 00:05:25.904 is increasing…the slope is increasing.And you 00:05:25.904 --> 00:05:28.844 see this over here, over there the slope is zero. The slope of the derivative is 00:05:28.844 --> 00:05:32.570 zero, the slope of the derivative self isn’t changing right this moment and then 00:05:32.570 --> 00:05:37.367 …and then you see that the slope is increasing. 00:05:37.367 --> 00:05:40.726 And once again we can visualize that on the second derivative, the derivative of the derivative. 00:05:40.726 --> 00:05:44.721 If the derivative is increasing that means the derivative of that must be positive. 00:05:44.721 --> 00:05:48.000 And it is indeed the case that the derivative is 00:05:48.000 --> 00:05:51.477 positive. And we have a word for this downward 00:05:51.477 --> 00:05:56.395 opening “U” and this upward opening “U”. we call this 00:05:56.395 --> 00:05:58.313 “ concave downwards” 00:05:58.313 --> 00:06:03.728 (let me make this clear)… 00:06:03.728 --> 00:06:07.995 concave downwards. 00:06:07.995 --> 00:06:11.708 And we call this “ concave upwards”… concave 00:06:11.708 --> 00:06:15.313 upwards. So let’s review how we can identify concave 00:06:15.313 --> 00:06:19.900 downwards intervals and upwards intervals. So we are talking about concave 00:06:19.900 --> 00:06:24.769 downwards…”concave downwards”. 00:06:24.769 --> 00:06:27.846 We see several things, 00:06:27.846 --> 00:06:32.769 we see that the slope is decreasing, the slope is 00:06:32.769 --> 00:06:37.172 is decreasing.“The slope 00:06:37.172 --> 00:06:40.999 is decreasing” which is another way of saying, 00:06:40.999 --> 00:06:42.647 which is another way of saying that f’(x) 00:06:42.647 --> 00:06:50.902 is decreasing. 00:06:50.902 --> 00:06:54.246 decreasing. Which is another way of saying that the 00:06:54.246 --> 00:06:57.573 second derivative must be negative. If the first derivative is decreasing, the second 00:06:57.573 --> 00:07:01.246 the second derivative must be negative. Which is another way 00:07:01.246 --> 00:07:04.902 of saying that the second derivative of that interval must 00:07:04.902 --> 00:07:09.307 be… must be negative. So if you have 00:07:09.307 --> 00:07:12.769 negative second derivative, then you are in a concave 00:07:12.769 --> 00:07:16.770 downward interval. Similarly…similarly 00:07:16.770 --> 00:07:20.393 (I have trouble saying that word), let’s think about concave upwards, 00:07:20.393 --> 00:07:25.244 where you have an upward opening “U”. Concave upwards. 00:07:25.244 --> 00:07:28.644 In these intervals, the slope is increasing, 00:07:28.644 --> 00:07:32.706 we have negative slope, less negative, less negative…zero, positive, more positive, more 00:07:32.706 --> 00:07:36.263 positive…even more positive. So slope...slope 00:07:36.263 --> 00:07:39.642 is increasing. "Slope is 00:07:39.642 --> 00:07:43.437 increasing which means 00:07:43.437 --> 00:07:48.041 that the derivative of the function is 00:07:48.041 --> 00:07:52.036 increasing. And you see that right over 00:07:52.036 --> 00:07:55.368 here, this derivative is increasing in value, 00:07:55.368 --> 00:07:59.263 which means that the second derivative，the second derivative 00:07:59.263 --> 00:08:03.433 over the interval where we are concave upwards must be greater than zero, 00:08:03.433 --> 00:08:07.440 the second derivative is greater than zero that means the first derivative is increasing, 00:08:07.440 --> 00:08:11.675 which means that the slope is increasing. We are in a concave upward, 00:08:11.675 --> 00:08:14.705 we are in a concave upward interval. Now, 00:08:14.705 --> 00:08:19.177 given all these definitions we’ve just given for concave downwards 00:08:19.177 --> 00:08:23.367 and concave upwards interval, can we come out with another way of indentifying whether a critical point 00:08:23.367 --> 00:08:26.770 is a minimum point or maximum point. 00:08:26.770 --> 00:08:30.595 Well, if you have a maximum point, if you have a critical point where the 00:08:30.595 --> 00:08:34.374 function...where the function is concave downwards, 00:08:34.374 --> 00:08:38.368 then it going to be a maximum point."Concave downwards". Let’s just be clear here, 00:08:38.368 --> 00:08:42.595 means that it’s opening down like this 00:08:42.595 --> 00:08:46.593 and we are talking about a critical point. If we’re assuming it’s concave downwards 00:08:46.593 --> 00:08:49.438 over here, we’re assuming differentiability over this interval and so the critical point 00:08:49.438 --> 00:08:53.318 is gonna be one where the slope is zero, so it’s gonna be that point 00:08:53.318 --> 00:08:57.770 right over there. So if you have a concave upwards and you have a point where 00:08:57.770 --> 00:09:01.866 f’(a) = 0 00:09:01.866 --> 00:09:04.872 then we have a maximum point at a. 00:09:04.872 --> 00:09:14.036 And similarly if we are a concave upwards 00:09:14.036 --> 00:09:17.707 that means that our function looks something like this and if we 00:09:17.707 --> 00:09:21.928 found the point. Obviously a critical point could also be where the function is not 00:09:21.928 --> 00:09:25.676 defined. But if we are assuming that our first derivative and second derivative is 00:09:25.676 --> 00:09:29.976 defined here then the critical point is going to be one where the first derivative is 00:09:29.976 --> 00:09:33.374 going to be zero, so f’(a) 00:09:33.374 --> 00:09:37.702 f’(a)= 0.If f’(a)= 0 00:09:37.702 --> 00:09:41.970 and if we are concave upwards and the interval around 00:09:41.970 --> 00:09:44.975 a, so the second derivative is greater than zero, then it’s pretty 00:09:44.975 --> 00:09:49.175 clear you see here that we are dealing with… we are dealing with a minimum, 00:09:49.175 --> 00:09:53.175 a minimum point at a .