Existence theorems intro | Existence theorems | AP Calculus AB | Khan Academy

What we're going to talk about in this video are three theorems that are sometimes collectively known as existence theorems.

So the first that we're going to talk about is the intermediate value theorem, and the common thread here, all of the existence theorems say, "Hey, we're looking for something over an interval. There exists an x value between A and B where something interesting happens."

In the intermediate value theorem, we assume that if we're continuous over the closed interval from A to B—and, in fact, all of these existence theorems assume that our function is continuous over the closed interval from A to B—then we take on every value between F of A and F of B.

Or another way to think about it is pick a value from F between F of A and F of B, including F of A or F of B. Let's call that value L. The intermediate value theorem tells us that if we make the assumption that f is continuous over this interval, then there must be a value between A and B that takes on the value L.

And I challenge you: try to draw a continuous function that goes from (A, F of A) to (B, F of B) that does not go through L. If you're continuous, you've got to go through L. The only way that you can't go through L—the other way—you could avoid going through L is if you are discontinuous. If you had a discontinuity right there, then you could avoid going through L.

But we're assuming we are continuous over the interval, and so here it's pretty intuitive that if you're continuous, there exists a c between A and B—including it could be A or B—that takes on any of those values. In this case, in this particular value that sits between F of A and F of B.

So that's the first existence theorem.

The second is what's often known as the extreme value theorem, and this one is similarly intuitive, I think. Once again, it assumes that f is continuous over A in this closed interval.

But here we say, look, if it is continuous over that closed interval, then there exist—that's why they're called existence theorems—there exist values between A and B, and it could happen at A or B where the function takes on a maximum, and there's a value between A and B where the function takes on a minimum value over that interval.

So once again, try to draw a function that does not take on a minimum and maximum value over that interval. If I just draw a straight line here, well, then the maximum value happens when x is B, and the minimum value happens when x is A.

If I do something like that, the maximum value occurs. Once again, it is occurring in this interval; it's occurring right here at C, and then the minimum value is occurring at A. If I draw something like—let me draw it like this—if I draw something like this, then the minimum value is occurring at this x value, and then the maximum value is occurring at that value, x value.

This is just saying that they exist; they exist over that interval, and it might happen at A; it might happen at B. And once again, the only way I can construct something where these maximum or minimum values, these extreme values, don't exist is if I make it discontinuous.

So for example, what if I had a graph that looked like—right at where we thought we were going to have a maximum value—we're discontinuous? Well, now you don't have a clear maximum value. Similarly, I could do something like this where now we do not have a clear minimum value, and so there does not exist an x where the function takes on a minimum value in that interval.

But once again, we are assuming that we are continuous, and so we will find these extreme values. And we have other videos where we go into much more depth on it.

Last but not least, to complete the trifecta, we have the mean value theorem, and this one is also intuitive. It starts going into differentiability and the derivative, and it adds an extra constraint for us above and beyond saying that we're continuous over the closed interval.

We also assume that we are differentiable over the open interval. If you're differentiable over an interval, it does mean that you're continuous, but if you're continuous, it does not necessarily mean that you're differentiable.

And all this tells us is if I have a continuous and differentiable function over this interval—differentiable over the open interval, continuous over the closed interval—so let me draw that.

So let me draw something like this, and if I were to compute the average rate of change from (A, F of A) to (B, F of B), the average rate of change I will do in this pink color. So that would be the slope of this line right over here; that would be the average rate of change, the slope of that line.

The mean value theorem tells us that there exists a point C where the derivative of our function at that point—the slope of the tangent line at that point—is the same as the average rate of change. We could eyeball that here; it looks like for this curve there's actually several points where the derivative looks the same as the average rate of change; maybe right over here, the slope of the tangent line looks like it has the exact same slope as the average rate of change.

So that could be our C; it exists. We feel good; it exists. What? There could be more than one value right over there; it looks like the slope is the same as the average rate of change. That could also be our C right over there.

So how could we construct something where this isn't true? Well, if we don't assume differentiability over the interval, we can actually find a continuous function where it isn't true, where you can find a point whose derivative is the same as the average rate of change.

So, for example, here's my countercase: maybe something like an absolute value function, where the average rate—let me draw a little bit better—so this was an A, B, some type of an absolute value function where this is linear up to this point and then linear up to this point. We're not going to be differentiable over here, so the derivative isn't defined at this point right over here.

Well now, you can't—at no point over this interval—is the slope of the tangent line the same as the slope connecting or is the same as the average rate of change. You might try to make an argument that, "Oh, maybe B," right over there, but we're not differentiable over there; there isn't a well-defined tangent line or—a well-defined tangent line and a well-defined derivative or slope of a tangent line.

So in other videos, we will go into more depth, but it's nice to look at them all together and see what they are all talking about. They're all talking about the existence of an x value in the interval where something interesting happens, where we take on a value between F of A and F of B, where we take on extreme values, or where the derivative at that point is the same as the average rate of change over the interval.