 Hi, I'm Zor. Welcome to a new Zor education. This is the lecture number four in the topic which I call main theorems in the derivative theory. All these theorems are called after certain famous mathematicians who introduce them first. Now today lecture is about theorem introduced by Cauchy, another French mathematician, and all these lectures are kind of increasing in complexity. Also, they're all very, actually, easy. All these theorems are very easy. But still they are kind of built upon each other. First there was a Fermat theorem, then there was a Rolls theorem, which is based on Fermat, then it was Lagrange theorem, which is based on the Roll. And this Cauchy is just a some kind of a little bit more complicated form of Lagrange. Also based, the proof is also based on the Rolls theorem. All right, so what is Cauchy's theorem? Here it is. So if you have two functions now, f of x and g of x both defined on a segment, a, b, with boundaries, then, what's important now, and that's the statement of the theorem is, there is such a point, x0, which belongs to this segment, where the ratio of derivatives at that point is equal to the ratio of increments of these functions on this segment. So this is increment of the function on this segment. This is increment of another function of this segment. So the ratio of derivatives is equal to the ratio. Now, obviously, when I'm saying something like this, I assume that this is not equal to 0, and this is not equal to 0. So all these considerations related to a little bit more rigorous proof, I assume as specified. So all I have to do is basically to prove this one. Now, as with the previous case in the Lagrange theorem, I will just introduce another function, which is a combination of these, and use the Roltz theorem, and from which I have this equation derived. Okay, so what's my auxiliary function? h of x is equal to f of x minus g of x times this ratio. Now, why is this function satisfies the conditions of the Roltz theorem? Well, let's just check what's the value of this function at both ends. At a, I have f of a minus g of a times this ratio. Okay, let's go to the common denominator. So, it's f of a times g of b minus f of a g of a minus g of a f of b and plus g of a f of a. Well, divided by g of b minus g of a. Okay, this and this cancel out. And what do I have? I have f a g b minus g a f b. Right, so let me do it here. f a g b minus g a f b divided by g of b minus f of a. And I wipe out this one. Okay, now h of b. Alright, it's f of b minus g of b times this ratio of increments equals. Okay, f of b g of b minus f of b g of a minus g of b f of b plus g of b f of a g b minus g a. So, what cancels out here? This. What remains? f a g b minus g a f b. Exactly the same thing. Alright? So, this is the same as this one. And if that is true, I can use the Rolls theorem for function g, for fashion h of x. So, we don't need this. All we need is just to take a particular derivative of this function and see what happens. Derivative is equal to, well, this is obviously linear combination of two functions. And that's derivative. Now, what the Rolls theorem says, that there is such a point at zero where this is equal to zero. From which, look at this, immediately follows this one. All you have to do is put this on the right and divide by derivative of g. Again, considering the derivative of g is not equal to zero at this point, etc., etc. So, under normal circumstances, when the function is smooth enough and we don't have zeros in the denominator, this is a true statement for some specific point x zero. It exists. So, the theorem says it exists. Point x zero exists, at least one point within this segment from a to b, where this particular ratio of derivatives is equal to the ratio of function increments on both ends. Well, basically, that's it. That's the last theorem which was named after some famous mathematician which I wanted to present to you. I do suggest you to go to websiteunisor.com where all these theorems are basically described. It's like in a textbook. Read it again. Try to consider the meaning of these and basically you will feel comfortable with all these theorems. I don't think it's very important to remember the names of which theorem is called after which mathematician. But in any case, I would say that the most important theorem is the first one, Fermat, which means if there is some kind of a local maximum or local minimum, then the derivative at this point is equal to zero because the tangential line is parallel to the x-axis. All right, so that's it for today. Thank you very much and good luck.