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What I have here in yellow is the graph of y=f（x）.

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That here in this move color I’ve graphed y’s equal

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to the derivative of f, is f′(x).And then here

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in blue I graphed y is equal to the second derivative of our function.

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So this is the derivative of this, of the first

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derivative right over there. And we’ve already seen examples of how

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can we identify minimum and maximum points. Obviously, if we have a graph in front of us,

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it’s not hard for human brain to identify this as a local

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maximum point. The function might take on higher values later on. And

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to identify this as a local minimum point. The function might take on

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lower values later on. But we saw, even if we don’t have a graph in front of

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us, if we are able to take the derivative of the function, we might…

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or if we are not able to take the derivative of the function. We might be able to identify

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these points as maximum or minimum. The way that we did it. Ok… what are the critical

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points for this function. Well, critical points over the function where the function’s

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derivative is either undefined or zero. This the function’s derivative.

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It’s zero here and here. So we would call those critical points.

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I don’t see any undefined. Any point was the derivative’s undefined

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just yet. So we would call here and

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here, critical points. So these are candidate

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minimum…these are candidate points which are function might take on a minimum or

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maximum value. And the way that we figured out whether it was a minimum or maximum

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value is to look at the behavior of the derivative around that point

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and over here we saw the derivative is de...or the

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derivative is positive.The derivative is positive

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as we approach that point

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and then it becomes negative. It goes from being positive

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to negative as we cross that point which means that the function]

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was increasing. If the derivative is positive that means the function was increasing

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as we approach that point and then decreasing as we leave that point.

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Which is a pretty good way to think about this… Being a maximum point,

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for increasing as we approach and decreasing as we leave it. Then this is definitely going

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to be a maximum point. Similarly,

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right over here, we see that the function is negative or the derivative

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is negative as we approach the point which means that the

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function is decreasing. And we see the derivative is

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positive as we exit that point. We go for having a negative derivative to a positive

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derivative which means the function goes from decreasing to

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increasing right around that point, which is a pretty good indication.

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Or that is an indication, that this critical point is a point at which the

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function takes on a minimum…a minimum value.

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What I want do now is to extend things by using

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the ideal of concavity… con-ca[ei]-vity.

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And I know I’m mispronouncing it, maybe it’s conca[æ]vity,

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but new thinking about concavity. Start to look at the second

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derivative, it rather than kind of seeing just as transition. To think about

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whether this is a minimum or maximum point. So

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let’s think about what’s happening in this first region. This kind of …this part of

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the curve up here where is it looks like an arc where it’s

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opening downward. Where it looks kinda like an “A” without the crossbeam or upside

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down “U” and then we’ll think about what’s happening in this kind of upward

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opening “U”, part of the curve. So over this first

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interval right over here, if we start we get this slope is very…is

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very ( actually I’ll do it in the same color, exactly the same color that

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I used for the actual derivative) the slope is very positive

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..slope is very positive. Then it becomes less positive...becomes

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less positive…then it becomes even less positive…becomes even less

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positive…and eventually gets to zero…eventually gets to zero. Then it keeps

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decreasing. Now becomes slightly negative…slightly negative. Then

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it becomes even more negative…becomes even more negative…and

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then it stops decreasing right around. It looks like it stops decreasing right

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around there. So the slope stops decreasing right around there. You see that in the red ,

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the slope is decreasing…decreasing…decreasing……until that point and

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then it starts to increase. So this entire section,

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this entire section right over here…

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the slope is decreasing. “Slope…

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slope is decreasing” and

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you see it right over here when we take the derivative, the deri…ative

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right over here… the entire, over this entire interval is decreasing.

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And we also see that when we take the second derivative. If the derivative is

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decreasing that means that the second, the derivative of the derivative is

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negative and we see that is indeed the case

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over this entire interval. The second derivative, the second

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derivative is indeed negative. Now what

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happens as we start to transition to this upward opening ”U” part of the curve.

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Well here the derivative is reasonably negative,

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it’s reasonably negative right there. But then it starts gets…it’s

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still negative but it becomes less negative and less negative

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…then it becomes zero,

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it becomes zero right over here. And then it becomes more and

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more and more positive, and you see that right over here. So over this

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entire interval, the slope or the derivative is increasing.

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So the slope...slope is

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is increasing…the slope is increasing.And you

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see this over here, over there the slope is zero. The slope of the derivative is

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zero, the slope of the derivative self isn’t changing right this moment and then

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…and then you see that the slope is increasing.

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And once again we can visualize that on the second derivative, the derivative of the derivative.

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If the derivative is increasing that means the derivative of that must be positive.

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And it is indeed the case that the derivative is

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positive. And we have a word for this downward

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opening “U” and this upward opening “U”. we call this

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“ concave downwards”

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(let me make this clear)…

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concave downwards.

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And we call this “ concave upwards”… concave

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upwards. So let’s review how we can identify concave

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downwards intervals and upwards intervals. So we are talking about concave

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downwards…”concave downwards”.

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We see several things,

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we see that the slope is decreasing, the slope is

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is decreasing.“The slope

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is decreasing” which is another way of saying,

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which is another way of saying that f’(x)

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is decreasing.

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decreasing. Which is another way of saying that the

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second derivative must be negative. If the first derivative is decreasing, the second

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the second derivative must be negative. Which is another way

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of saying that the second derivative of that interval must

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be… must be negative. So if you have

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negative second derivative, then you are in a concave

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downward interval. Similarly…similarly

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(I have trouble saying that word), let’s think about concave upwards,

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where you have an upward opening “U”. Concave upwards.

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In these intervals, the slope is increasing,

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we have negative slope, less negative, less negative…zero, positive, more positive, more

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positive…even more positive. So slope...slope

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is increasing. "Slope is

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increasing which means

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that the derivative of the function is

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increasing. And you see that right over

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here, this derivative is increasing in value,

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which means that the second derivative，the second derivative

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over the interval where we are concave upwards must be greater than zero,

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the second derivative is greater than zero that means the first derivative is increasing,

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which means that the slope is increasing. We are in a concave upward,

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we are in a concave upward interval. Now,

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given all these definitions we’ve just given for concave downwards

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and concave upwards interval, can we come out with another way of indentifying whether a critical point

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is a minimum point or maximum point.

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Well, if you have a maximum point, if you have a critical point where the

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function...where the function is concave downwards,

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then it going to be a maximum point."Concave downwards". Let’s just be clear here,

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means that it’s opening down like this

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and we are talking about a critical point. If we’re assuming it’s concave downwards

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over here, we’re assuming differentiability over this interval and so the critical point

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is gonna be one where the slope is zero, so it’s gonna be that point

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right over there. So if you have a concave upwards and you have a point where

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f’(a) = 0

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then we have a maximum point at a.

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And similarly if we are a concave upwards

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that means that our function looks something like this and if we

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found the point. Obviously a critical point could also be where the function is not

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defined. But if we are assuming that our first derivative and second derivative is

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defined here then the critical point is going to be one where the first derivative is

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going to be zero, so f’(a)

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f’(a)= 0.If f’(a)= 0

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and if we are concave upwards and the interval around

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a, so the second derivative is greater than zero, then it’s pretty

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clear you see here that we are dealing with… we are dealing with a minimum,

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a minimum point at a .