The Help I need I am preparing for a competitive math exam. I need to learn methods of solving questions like this

If f$\left(\frac{x+y}{3}\right)= \frac{2+f(x)+f(y)}{3}$ for all real $x$ and $y$ and $f'(2)=2$, then determine y=f(x)

My Approach I can not approach because i haven't studied enough .That's why i need book suggestions I don't want to study functional equations deeply right now. I want study just enough to solve these kind of (elementary level)problems as given above.

Please Suggest me some appropiate books


closed as off-topic by user21820, B. Mehta, Henrik, Jack, Namaste Nov 30 '17 at 18:45

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Roughly speaking, there are no methods. Functional equations are mostly a dead end. The lucky few that can be solved are usually dealt with by repeated plugging of various suitable values as arguments.

To begin with, let's get rid of 2, for I feel it would look a little simpler that way: $$f(x)=g(x)+2\text{, then}\\ g\left({x+y\over3}\right)={g(x)+g(y)\over3} $$ Now plug $0$ for both arguments, and you'll find out that $g(0)=0$.

Now plug $x$ and $2x$, and you'll see that $g(2x)=2g(x)$.

At this point we might start to suspect that our equation is in fact equivalent to Cauchy's functional equation. We may continue in this manner until our domain is shredded according to Hamel basis. If you don't like the taste of it, then skip this step. Anyway, one particular family of solutions was already obvious before it: $g(x)=kx$, or $\color{red}{f(x)=kx+2}$. (Plug it into your original equation to make sure it fits!)

Now, if you want your derivative anywhere to be 2, then $k=2$ and we are done.

So it goes.

  • $\begingroup$ Please suggest me a book according to my purpose $\endgroup$ – Kislay Tripathi Oct 26 '17 at 12:57

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