Intermediate and extreme value theorems
Existence theorems intro
- [Instructor] 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 there, 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, comma f of a, to b, comma 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, the only you could avoid going through l is if you were discontinuous, if you had a discontinuity right there, then you could avoid going through l, but we're assuming that we're 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, and in this case, 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 comma, over this closed interval. But here, we say look, if it is continuous over that closed interval, then there exists, that's why they're called existence theorems, there exists 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, 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 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 gonna 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 differentiatability in 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're 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 comma f of a to b comma f of b, so 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. And 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. But there could 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 that 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 counter case. Maybe something like an absolute value function, where the average rate, where the average rate of change, let me draw it a little bit better. So this was 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 not, 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 conn--, or is the same as the average rate of change. You might try to make an argument that oh, maybe right over there, but we're not differentiable over there. There isn't a well-defined tangent line, or there isn't a well-defined tangent line and a well-defined derivative or slope of a tangent line. So in other videos we will go into depth. But it's nice to look at them all together. 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.