TIFR-2018-Maths-B: 5

Question

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

• $f$ has derivatives of all orders, and
• for all $x,y \in \mathbb{R}$,

$$f\left ( x+y \right )-f\left ( y-x \right )=2x{f}'\left ( y \right ).$$

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. There exists $f \in A$ which is not a polynomial.
• D. There exists $f \in A$ which is a polynomial of degree $4$.