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

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?

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 $$\le 2$$.
• Thanks. But, there may be a typing error, it should $y \leftarrow \frac{t}{2}$ – Offlaw Nov 26 '18 at 5:21