 If we know f is differentiable over an interval, we know quite a bit more about its behavior over that interval. Suppose f is differentiable over some interval between a and b, then we know... well, let's see what we know. So for example, suppose f' of x is continuous over the interval between 0 and 5, must f of x be continuous at x equals 0. And so we might say something like, well, we know the differentiability requires continuity, so we must be continuous at the endpoints. But let's be a little bit more careful about this, and the thing to remember is that if you don't find the flaws in your reasoning, someone else will. Now, in order for f to be continuous at x equals 0, then we know the limit and the function value have to exist. So if f of 0 isn't defined, then f won't be continuous. Well, a simple function for which f of 0 is undefined is f of x equals 1 divided by x, and we see that the derivative is equal to... and this is defined and continuous over the interval between 0 and 5, as long as we exclude x equals 0. So it's possible for a function to be differentiable over an interval, but not continuous at the endpoints. So this leads to two useful results. Actually, one pre-result and one actually useful result. The pre-result is something called Rohl's theorem. This is named after a French mathematician by the name of Michel. I can't remember his last name. I'll think of it in a moment. Anyway, he discovered the following. Suppose our function f satisfies the following. f of a is equal to f of b. f is continuous over the closed interval between a and b, that includes both a and b, and f is differentiable over the open interval from a to b, which does not necessarily include x equals a or x equals b. Then there's some c in this interval and the derivative of c is equal to 0. And one important thing to note about Rohl's theorem is it's what mathematicians call an existence theorem. It's saying that something exists, but it doesn't tell you how to find it. And so we don't know the actual value of c, only that it must exist. And it's also worth reading the fine print, which in this case doesn't exist. In particular, while this says there's some c, it doesn't preclude the possibility that there may be more than 1. So there could be more than one place where our derivative is equal to 0. Let's build up a little bit of intuition for what Rohl's theorem tells us. Since f of x is continuous, and f of a equals f of b, the graph of y equals f of x starts and ends at the same level, that's this equality of the function values of our endpoints, with no breaks. That's our continuity requirement. And since f prime of x exists for all x in the interval, there are no cusps or points of non-differentiability. We always have a tangent line. And since Rohl's theorem claims that there is some place where f prime of c is equal to 0, this says that somewhere there's a place where the tangent line is horizontal. Now, Rohl's theorem by itself isn't all that useful because the conditions we have to have in order for it to work are a little bit restrictive. We have to start and end at the same function value. But it does give us a much more important result known as the mean value theorem. And we'll start off in the very similar way. Suppose f is continuous over a closed interval and differentiable over an open interval. Then there's some c in that interval where the derivative is equal to this expression. Now, that's a weird expression. I guess it uses the beginning and the ending points, but what does it actually mean? Well, let's see if we can make sense out of this. So we have a function that is continuous and differentiable. Let's take a look at the graph. So our interval runs from x equals a to x equals b. And so let's graph our function. We start somewhere, we end somewhere, and we're continuous between the two points. And remember, if it's not written down, it didn't happen. Let's go ahead and label these points and the graph. And if we sit and stare at this, we realize that the value of this expression is the slope of the straight line between the end points on the graph of y equals f of x. So the mean value theorem says that there's some point where the tangent line parallels the straight line. So for example, suppose my function is continuous over the closed interval between 5 and 10 and differentiable over the open interval between 5 and 10. If we know where we are at 5 and at 10, what do we know about both our function and our derivative in the interval? Well, since this is in the video about the mean value theorem, you could probably guess that the mean value theorem is applicable. But again, to get a feel for what this really means, let's go ahead and use this information to construct a graph. We know the function values at 5 and at 10. So those give us two points on the graph. And again, if it's not written down, it didn't happen. Label. f of x is continuous and differentiable, so there's no cusps or corners, and it's an unbroken graph. And so the mean value theorem guarantees there is some c for which the derivative at c is equal to the slope of the straight line connecting the endpoints. And that slope will be... But wait, there's more. We can find some more about this. We know that f of x is continuous over the interval, and so the intermediate value theorem tells us that for any value m between f of 5, 10 and f of 10, negative 3, there is some value d for which f of d is equal to m. And, well, this is positive, this is negative, so we know that f of x equals 0 has some solution x equals d, where d is between 5 and 10.