AP®︎ Calculus BC (2017 edition)
Sal introduces the "critical points" of a function and discusses their relationship with the extremum points of the function. Created by Sal Khan.
I've drawn a crazy looking function here in yellow. And what I want to think about is when this function takes on the maximum values and minimum values. And for the sake of this video, we can assume that the graph of this function just keeps getting lower and lower and lower as x becomes more and more negative, and lower and lower and lower as x goes beyond the interval that I've depicted right over here. So what is the maximum value that this function takes on? Well we can eyeball that. It looks like it's at that point right over there. So we would call this a global maximum. the? Function never takes on a value larger than this. So we could say that we have a global maximum at the point x0. Because f of of x0 is greater than, or equal to, f of x, for any other x in the domain. And that's pretty obvious, when you look at it like this. Now do we have a global minimum point, the way that I've drawn it? Well, no. This function can take an arbitrarily negative values. It approaches negative infinity as x approaches negative infinity. It approaches negative infinity as x approaches positive infinity. So we have-- let me write this down-- we have no global minimum. Now let me ask you a question. Do we have local minima or local maxima? When I say minima, it's just the plural of minimum. And maxima is just the plural of maximum. So do we have a local minima here, or local minimum here? Well, a local minimum, you could imagine means that that value of the function at that point is lower than the points around it. So right over here, it looks like we have a local minimum. And I'm not giving a very rigorous definition here. But one way to think about it is, we can say that we have a local minimum point at x1, as if we have a region around x1, where f of x1 is less than an f of x for any x in this region right over here. And it's pretty easy to eyeball, too. This is a low point for any of the values of f around it, right over there. Now do we have any other local minima? Well it doesn't look like we do. Now what about local maxima? Well this one right over here-- let me do it in purple, I don't want to get people confused, actually let me do it in this color-- this point right over here looks like a local maximum. Not lox, that would have to deal with salmon. Local maximum, right over there. So we could say at the point x1, or sorry, at the point x2, we have a local maximum point at x2. Because f of x2 is larger than f of x for any x around a neighborhood around x2. I'm not being very rigorous. But you can see it just by looking at it. So that's fair enough. We've identified all of the maxima and minima, often called the extrema, for this function. Now how can we identify those, if we knew something about the derivative of the function? Well, let's look at the derivative at each of these points. So at this first point, right over here, if I were to try to visualize the tangent line-- let me do that in a better color than brown. If I were to try to visualize the tangent line, it would look something like that. So the slope here is 0. So we would say that f prime of x0 is equal to 0. The slope of the tangent line at this point is 0. What about over here? Well, once again, the tangent line would look something like that. So once again, we would say f prime at x1 is equal to 0. What about over here? Well, here the tangent line is actually not well defined. We have a positive slope going into it, and then it immediately jumps to being a negative slope. So over here, f prime of x2 is not defined. Let me just write undefined. So we have an interesting-- and once again, I'm not rigorously proving it to you, I just want you to get the intuition here. We see that if we have some type of an extrema-- and we're not talking about when x is at an endpoint of an interval, just to be clear what I'm talking about when I'm talking about x as an endpoint of an interval. We're saying, let's say that the function is where you have an interval from there. So let's say a function starts right over there, and then keeps going. This would be a maximum point, but it would be an end point. We're not talking about endpoints right now. We're talking about when we have points in between, or when our interval is infinite. So we're not talking about points like that, or points like this. We're talking about the points in between. So if you have a point inside of an interval, it's going to be a minimum or maximum. And we see the intuition here. If you have-- so non-endpoint min or max at, let's say, x is equal to a. So if you know that you have a minimum or a maximum point, at some point x is equal to a, and x isn't the endpoint of some interval, this tells you something interesting. Or at least we have the intuition. We see that the derivative at x is equal to a is going to be equal to 0. Or the derivative at x is equal to a is going to be undefined. And we see that in each of these cases. Derivative is 0, derivative is 0, derivative is undefined. And we have a word for these points where the derivative is either 0, or the derivative is undefined. We called them critical points. So for the sake of this function, the critical points are, we could include x sub 0, we could include x sub 1. At x sub 0 and x sub 1, the derivative is 0. And x sub 2, where the function is undefined. Now, so if we have a non-endpoint minimum or maximum point, then it's going to be a critical point. But can we say it the other way around? If we find a critical point, where the derivative is 0, or the derivative is undefined, is that going to be a maximum or minimum point? And to think about that, let's imagine this point right over here. So let's call this x sub 3. If we look at the tangent line right over here, if we look at the slope right over here, it looks like f prime of x sub 3 is equal to 0. So based on our definition of critical point, x sub 3 would also be a critical point. But it does not appear to be a minimum or a maximum point. So a minimum or maximum point that's not an endpoint, it's definitely going to be a critical point. But being a critical point by itself does not mean you're at a minimum or maximum point. So just to be clear that all of these points were at a minimum or maximum point. This were at a critical point, all of these are critical points. But this is not a minimum or maximum point. In the next video, we'll start to think about how you can differentiate, or how you can tell, whether you have a minimum or maximum at a critical point.