 Hi, I'm Zor. Welcome to Unizor Education. I would like to continue talking about different important theorems, which have certain names of mathematicians who proved them. The first one, the previous lecture, was about Pierre-Fermas theorem. Now this one is called the Rolls theorem, that's the name of the guy. So, what is all about? Now I will continue using the same terminology as I introduced in the previous lecture, which means function is supposed to be smooth, and then we'll talk about its derivative. So, here is what Rolls theorem is about. Let's assume you have a function which takes the same values on both ends of certain segment. Now, in this case, I'm talking about segment, which means ends are included. So, we're not talking about infinity, we're talking about a finite segment. I mean, maybe function is defined somewhere else, doesn't really matter right now. But we're talking about concrete segment from A to B, where B is greater than A, and the value of the function is the same. F of A is equal to F of B. And the function is smooth enough, which means it has derivative, derivative is continuous, maybe it has a second derivative, whatever, smooth enough. So, for this smooth function, what this theorem actually states is that there is some point somewhere between X, between A and B, X0, where the function is reaching its local maximum, at least one point, maybe more than one point, because maybe the function looks like this. Again, the same values on both ends, but in this particular case, function has one, two, three, four different local extremums, two maximum and two minimums, right? But what I'm talking about that there is at least one such point, call it X0, where the function's derivative is equal to zero and it means it's reaching its local maximum or minimum, local extremum. Okay? All right, how can I prove it? Well, let's just think about it. If the function's values are the same on both ends, then the function either is constant and it's always equal some value, f at A or f at B. Now, for a constant, we know that the derivative of the function is equal to zero everywhere, right? So, our theorem is proven in this particular point. Any point within this interval from A to B is the point where derivative is equal to zero. Or the function is not constant. Well, which means it goes up or down from f at A, up or down, up or down, up or down, and then eventually it comes to f of B. Which means that the function should be monotonically increasing and then decreasing. Otherwise, it would not come to the same value. If the function is monotonically increasing, then this value would be greater than this one. So, it's not monotonically increasing. Which means after increase, you will have decrease. Or after decrease, it will be increase. So, considering that the derivative is a continuous function, if the function is increasing, derivative is positive, the function decreasing derivative is negative. So, somewhere it should cross zero. And that's basically the point we are talking about. So, at least one point where the sign of the derivative is changing from plus to minus or from minus to plus is the sign. It is exactly the point where the derivative must be equal to zero. Which means it's a local extremum according to the previous theorem Fermat, which we have proven in the previous lecture. That's basically it. It's a very simple thing. So, all you have to do is just logically consider that the function which takes the same values on both ends must be either constant or after increasing, it should decrease. Or after decreasing, it should increase. Maybe more than once, but at least once it must be done. Okay, that's it. I do suggest you to read the notes for this lecture. They're basically the same as I was just saying. Just, you know, in case you would like to read it as a textbook, if you wish. That's it for this particular theorem. Thank you very much and good luck.