Suppose $ f $ is an odd function and is differentiable everywhere. Prove that for every positive number $ b $, there exists a number $ c $ in $ (-b, b) $ such that $ f'(c) = f(b)/b $.
So we are being a staff supposed f isn't art function and it's defensible everywhere. Proved that for every number, for every positive number B, there exist a number C and negative deeds to be such that at primacy as you could ever be Overby. So we are. There's a couple of things we need to take a note here. So our interval is actually negative b to be and there's something really important. And they told us that this function is differential everywhere. If they tell us this right away, we can also if it's defensible, we also have We can assume that it's continuous because in order for a function to be defensible, it has to be continuous. So this follows mean value still himself follows means valued well mean MBT So we know that we're going to have to use the mean Barry threw him at some point. And since we have our two like points, we can actually go ahead And you applied to mean very serum in this Ah, in this interval so we could say that there exists of C, so start the derivative at six. Just the average for between B and negative B so this would be f O B minus half of negative B all over B minus negative B. So all I did was I just rewrote there the mean very serum in terms of the point negative B to B. And now, before we move on, this tells a very, very important piece of information in here to tell us that F is an odd function. Now, if you recall our functions, you know, out of function by putting in and negative inside the function and the definition not function gives you a negative out front. And this and this you can learn Look back at it through a graphically because there are some functions like the Cuban, I mean the cubic function. So this is not a very good trying. Just do this again. I'll jot down here just to remind you of what the Cuba function is. So the point here one I mean negative one here will give us the same value as is at the negative point, which would be right here. So that's what that's what. And this is an art function. So this is all an odd function is it's just a statement This is how you know it's not fun. But since this is a non function, we can actually rewrite this statement right here. So we're going to rewrite it on the right. Prime of C is equal to after b minus negative F b. And then this would be to be because two makes a positive. And then if you look closely, this is a negative negative. So this makes a positive and a positive, and then this gives us a look to the next page. This gives us f primacy equal two f b over to B under two cancels, and we have our statement that we want does. You're supposed to be proving you remember a promise e f a b o b. We have complete the truth.