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小萨介绍了凹凸性的概念,一个图像“向上凹”或“向下凸”的意思以及凹性和二阶导的联系。 Sal Khan 创建




What I have here in yellow is the graph of y=f(x). That here in this move color I’ve graphed y’s equal to the derivative of f, is f′(x).And then here in blue I graphed y is equal to the second derivative of our function. So this is the derivative of this, of the first derivative right over there. And we’ve already seen examples of how can we identify minimum and maximum points. Obviously, if we have a graph in front of us, it’s not hard for human brain to identify this as a local maximum point. The function might take on higher values later on. And to identify this as a local minimum point. The function might take on lower values later on. But we saw, even if we don’t have a graph in front of us, if we are able to take the derivative of the function, we might… or if we are not able to take the derivative of the function. We might be able to identify these points as maximum or minimum. The way that we did it. Ok… what are the critical points for this function. Well, critical points over the function where the function’s derivative is either undefined or zero. This the function’s derivative. It’s zero here and here. So we would call those critical points. I don’t see any undefined. Any point was the derivative’s undefined just yet. So we would call here and here, critical points. So these are candidate minimum…these are candidate points which are function might take on a minimum or maximum value. And the way that we figured out whether it was a minimum or maximum value is to look at the behavior of the derivative around that point and over here we saw the derivative is de...or the derivative is positive.The derivative is positive as we approach that point and then it becomes negative. It goes from being positive to negative as we cross that point which means that the function] was increasing. If the derivative is positive that means the function was increasing as we approach that point and then decreasing as we leave that point. Which is a pretty good way to think about this… Being a maximum point, for increasing as we approach and decreasing as we leave it. Then this is definitely going to be a maximum point. Similarly, right over here, we see that the function is negative or the derivative is negative as we approach the point which means that the function is decreasing. And we see the derivative is positive as we exit that point. We go for having a negative derivative to a positive derivative which means the function goes from decreasing to increasing right around that point, which is a pretty good indication. Or that is an indication, that this critical point is a point at which the function takes on a minimum…a minimum value. What I want do now is to extend things by using the ideal of concavity… con-ca[ei]-vity. And I know I’m mispronouncing it, maybe it’s conca[æ]vity, but new thinking about concavity. Start to look at the second derivative, it rather than kind of seeing just as transition. To think about whether this is a minimum or maximum point. So let’s think about what’s happening in this first region. This kind of …this part of the curve up here where is it looks like an arc where it’s opening downward. Where it looks kinda like an “A” without the crossbeam or upside down “U” and then we’ll think about what’s happening in this kind of upward opening “U”, part of the curve. So over this first interval right over here, if we start we get this slope is very…is very ( actually I’ll do it in the same color, exactly the same color that I used for the actual derivative) the slope is very positive ..slope is very positive. Then it becomes less positive...becomes less positive…then it becomes even less positive…becomes even less positive…and eventually gets to zero…eventually gets to zero. Then it keeps decreasing. Now becomes slightly negative…slightly negative. Then it becomes even more negative…becomes even more negative…and then it stops decreasing right around. It looks like it stops decreasing right around there. So the slope stops decreasing right around there. You see that in the red , the slope is decreasing…decreasing…decreasing……until that point and then it starts to increase. So this entire section, this entire section right over here… the slope is decreasing. “Slope… slope is decreasing” and you see it right over here when we take the derivative, the deri…ative right over here… the entire, over this entire interval is decreasing. And we also see that when we take the second derivative. If the derivative is decreasing that means that the second, the derivative of the derivative is negative and we see that is indeed the case over this entire interval. The second derivative, the second derivative is indeed negative. Now what happens as we start to transition to this upward opening ”U” part of the curve. Well here the derivative is reasonably negative, it’s reasonably negative right there. But then it starts gets…it’s still negative but it becomes less negative and less negative …then it becomes zero, it becomes zero right over here. And then it becomes more and more and more positive, and you see that right over here. So over this entire interval, the slope or the derivative is increasing. So the slope...slope is is increasing…the slope is increasing.And you see this over here, over there the slope is zero. The slope of the derivative is zero, the slope of the derivative self isn’t changing right this moment and then …and then you see that the slope is increasing. And once again we can visualize that on the second derivative, the derivative of the derivative. If the derivative is increasing that means the derivative of that must be positive. And it is indeed the case that the derivative is positive. And we have a word for this downward opening “U” and this upward opening “U”. we call this “ concave downwards” (let me make this clear)… concave downwards. And we call this “ concave upwards”… concave upwards. So let’s review how we can identify concave downwards intervals and upwards intervals. So we are talking about concave downwards…”concave downwards”. We see several things, we see that the slope is decreasing, the slope is is decreasing.“The slope is decreasing” which is another way of saying, which is another way of saying that f’(x) is decreasing. decreasing. Which is another way of saying that the second derivative must be negative. If the first derivative is decreasing, the second the second derivative must be negative. Which is another way of saying that the second derivative of that interval must be… must be negative. So if you have negative second derivative, then you are in a concave downward interval. Similarly…similarly (I have trouble saying that word), let’s think about concave upwards, where you have an upward opening “U”. Concave upwards. In these intervals, the slope is increasing, we have negative slope, less negative, less negative…zero, positive, more positive, more positive…even more positive. So slope...slope is increasing. "Slope is increasing which means that the derivative of the function is increasing. And you see that right over here, this derivative is increasing in value, which means that the second derivative,the second derivative over the interval where we are concave upwards must be greater than zero, the second derivative is greater than zero that means the first derivative is increasing, which means that the slope is increasing. We are in a concave upward, we are in a concave upward interval. Now, given all these definitions we’ve just given for concave downwards and concave upwards interval, can we come out with another way of indentifying whether a critical point is a minimum point or maximum point. Well, if you have a maximum point, if you have a critical point where the function...where the function is concave downwards, then it going to be a maximum point."Concave downwards". Let’s just be clear here, means that it’s opening down like this and we are talking about a critical point. If we’re assuming it’s concave downwards over here, we’re assuming differentiability over this interval and so the critical point is gonna be one where the slope is zero, so it’s gonna be that point right over there. So if you have a concave upwards and you have a point where f’(a) = 0 then we have a maximum point at a. And similarly if we are a concave upwards that means that our function looks something like this and if we found the point. Obviously a critical point could also be where the function is not defined. But if we are assuming that our first derivative and second derivative is defined here then the critical point is going to be one where the first derivative is going to be zero, so f’(a) f’(a)= 0.If f’(a)= 0 and if we are concave upwards and the interval around a, so the second derivative is greater than zero, then it’s pretty clear you see here that we are dealing with… we are dealing with a minimum, a minimum point at a .