 Hi, I'm Zor. Welcome to Inizor Education. I would like to start a small set of lectures related to named theorems about derivatives, named in terms of one theorem is called Fermat theorem, another was Lagrange, another was Rohl, another was Cauchy. I mean there are different mathematicians who actually contributed to all these theorems. They're all very, very simple and probably like 200, 300 years ago when they were actually first introduced and proven by these people and some other people. So it's quite interesting actually their fundamental theorems for derivatives and they're very, very useful quite frankly. So today's lecture will be about the Fermat theorems. Fermat theorem only. Now this lecture and all others are part of the course of advanced mathematics for high school students and teenagers. The course is presented on Unizor.com website. So if you're just watching it from YouTube I do recommend you to go from Unizor.com because every lecture has very detailed description plus there is certain educational functionality. You can take exam for instance if you want to. The site is completely free. So we'll talk about Fermat theorem. Fermat theorem is actually about internal local extremum of the function. So let me just explain terminology here. First of all what does it mean internal? Well we are talking about function which is defined on certain interval. Function interval can be infinite as well. And internal it means that we are talking about the characteristics of this function inside this interval around certain points. These points will be called later on station and I'll explain why. So internal means it's the behavior not an entire interval where it's defined but locally at certain points. Now then we are talking about local behavior of the function which means we are talking about the behavior of the function at certain points and in immediate vicinity locality around that point. And the third word which I was using so internal local extremum. Now extremum is actually one word which means minimum or maximum. Well this is local minimum and this is local maximum. So you see this is where locality actually the property of locality is concentrated. So what does it mean that this is a local maximum? Well it means that in immediate neighborhood of this point the function is less than its value at that particular point. Local minimum is where the function in the immediate vicinity of this point is greater than the value of this point. So that's what basically it means. So this is about the behavior of the function. Now about the quality of the function itself. We are talking about functions which are differentiable. Which in particular actually from differentiability follows its continuousness continuity. And also what's also important the derivative itself should also be relatively smooth. Which means at least continuous or maybe in some cases I would require that the derivative is differentiable itself. So it's like a second derivative exists. So all these properties of the function I would call a smoothness of the function. If the function is smooth well it means basically it has all the features sufficient for whatever the logic I will use to be basically valid. So sometimes smoothness just differentiability. Sometimes it means that derivative not only exists but also it's supposed to be a continuous function. In some other cases derivative might must be differentiable itself. So it doesn't really matter right now. I don't want to be very rigorous in my definition of the smoothness of the function. But I think that from context you would probably feel what it is. Now this is a smooth function. Now this function is not smooth. You see this angle that's where it's not differentiable. So we are talking about smooth functions and whatever statements I will make I will make only about these smooth functions. Alright so what is the Fermat theorem? Fermat theorem is the following. If the function has local extremum this or this then its derivative is supposed to be zero at this point. Now you remember what derivative geometrically means. If you have something like this which is line tangential to our function then derivative basically means tangent of this angle. So if the function is having maximum or minimum intuitively it's kind of obvious that the tangential line should be horizontal here and horizontal here. So intuitively it's obvious that the angle is supposed to be equal to zero two parallel lines so the angle between them is zero. So tangent of this is zero so derivative is equal to zero at this particular point. Let's consider this one. It doesn't really matter. Okay now how can I prove it? First of all from the intuitive standpoint it's obvious, geometrically it's kind of obvious too. Well let me just talk a little bit about just common sense basically. Look if this is local minimum for instance it means that the function is decreasing monotonically decreasing before and monotonically increasing after this point right? I'm not talking about everywhere, no in the immediate vicinity in the neighborhood of this point. So it may be very small neighborhood but it must be some neighborhood where the function is monotonically decreasing before this point and increasing after. And we know that monotonically decreasing point have negative derivative. Monotonically increasing functions have positive derivative. So what happens is the derivative is negative negative negative negative here. Now I don't know what's in this particular point but I do know that after this point it becomes positive. So something is negative then something is positive my function is smooth enough in this particular case it means derivative is a continuous function then there should be crossing between negative and positive should be crossing zero right? Otherwise the function derivative would not be continuous function. So that's exactly the point where we are crossing from negative to positive which means it must be equal to zero. On another hand maybe a little bit more vigorous I don't know. Let's consider that the function is not, the derivative of the function is not equal to zero at this particular point. Well if it's not equal to zero then it's either positive or negative right? Now if it's positive it means as we have proven before the function should be monotonically increasing at this particular point. Well if function is monotonically increasing at certain point then this point cannot be local minimum because it means that before it was even smaller right? Same thing it cannot be monotonically increasing, it cannot be monotonically decreasing which means that the derivative cannot be positive nor negative. It must be equal to zero. So what our assumptions are in this case? Well the function is smooth enough to make all this logic actually working. That's all. I don't want to go into really vigorous definition of what is exactly smooth function but in this case you understand what it is. Now absolutely similar with the local maximum. So from more intuitive standpoint we have a derivative which is changing from the positive to a negative. So at this point it must be equal to zero. It must cross the point zero or you can make some other reasoning for instance what if it's not zero at this particular point then it should be either negative or positive in which case if it's positive then it would be greater to the right of this function, of this point so it cannot be local maximum. If it's negative then on the left it would be greater. So in any case there would be no local maximum. Well basically that's it. My whole proof was very very simple as you see but I wanted to introduce you to these concepts which are very important for derivatives. Well first most important kind of result of this particular experiment with the mass logic is that if you have local maximum or minimum then the tangential line, this horizontal derivative is equal to zero. What is another important point is a concept of a smooth function sufficiently smooth to make the proof actually relatively rigorous. Because we don't really want any kind of special cases like this. What happens if this is the local minimum when the function is not differentiable. We are not considering these cases. The cases which we are considering are all related to smooth function and all the function which you will be dealing with like polynomial functions exponential functions, logarithm functions, trigonometry trigonometry. All these functions are smooth enough. Actually they are differentiable any number of times. Any higher order derivative exists for these functions. Okay, so basically that's it. That's all I wanted to say about the Fermat's theorem. By the way when you're saying Fermat's theorem it's not exactly one particular theorem. There are many Fermat's theorem. This is one of the Fermat's theorem. It's a theorem about equality of the derivative to zero in extremum points. Points of extremum. Okay, I just suggested to read the notes for this lecture. They are basically more or less the same as whatever I'm saying right now. But if you read it again it just probably better understood if you wish. Okay, that's it. Thank you very much and good luck.