I am solving previous year question paper some competitive exam. Give me some hint to solve the following problem.
Let $f$ be an entire function. Suppose for each $a \in \mathbb{R} $ there exists at least one coefficient $c_n$ in $f(z) = \sum_{n=0}^{\infty} c_n (z-a)^n$ which is zero. Then,
a) $f^{(n)}(0) = 0$ for infinitely many $n \ge 0$
b) $f^{(2n)}(0)=0$ $ \forall n \ge 0$
c) $f^{(2n+1)}(0)=0$ $ \forall n \ge 0$
d) $f^{(n)}(0) = 0$ for all sufficiently large n.
Thanks in advance.
$f$ is a polynomial and d) is correct.
If $f$ is not a polynomial, then $f^{(n)}$ is a non constant entire function for all $n$. The set of zeros of $f^{(n)}$ is discrete, and in particular, countable. Thus, $$ Z=\{z\in\mathbb{C}:f^{(n)}(z)=0\text{ for some }n\} $$ is countable. But $\mathbb{R}\subset Z$, a contradiction.