Let $A$ be the set of all functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy the following two properties.

Question

Let $A$ be the set of all functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy the following two properties:

(1) $f$ has derivative of all orders, and

(2) for all $x,y \in \mathbb{R}$, $f(x+y)-f(y-x)=2xf'(y)$.

Which of the following sentences is true?

(a) Any $f\in A$ is a polynomial of degree less than or equal to 1

(b) Any $f \in A$ is a polynomial of degree less than or equal to 2

(c) $\exists f \in A$ which is not polynomial

(d) $\exists f \in A$ which is a polynomial of degree 4

It is a problem from TIFR GS-2018. I have found polynomials upto degree 2 hold this properties but have to exclude the possibility of (c) and (d). I also found out if $f'(x)$ has at most one real root and clearly graph of $f$ is symetric about that root. How can I proceed further?

Answer 1

We have $$f(x+y)-f(y-x)=2xf'(y) $$ Apply $\frac{\mathrm d}{\mathrm dx}$: $$f'(x+y)+f'(y-x)=2f'(y) $$ and once more: $$f''(x+y)-f''(y-x)=0. $$ With $x\leftarrow \frac t2$, $t\leftarrow \frac t2$, this becomes $$f''(t)=f''(0). $$ Thus $f$ is a polynomal and of degree $\le 2$.