Q:

Number of zeros of $ f^n $

Let $f:\Bbb R\to \Bbb R$ be infinetly differentiable function that vanishes at $10$ distinct points in $\Bbb R$.suppose $ f^{n} $ denote $n$-th derivate of $f$, for $n \ge 1$. Then which of following is true.

$ f^n $ has atleast $10$ zeros, for $1\le n\le8$
$ f^n $ has atleast $1$ zero, for $1\le n\le9$
$ f^n $ has atleast $10$ zeros, for $ n\ge10$
$ f^n $ has atleast $1$ zeros, for $ n\ge9$

I am still finding a way to begin.correct answer is $2$

A:

Hint: if $f(x) = 0$ for $10$ distinct $x \in \Bbb R$, how many zeros must $f'(x)$ have by Rolle's theorem?  If $f'(x)$ has at least that number of zeros, how many zeros must $f''(x)$ have?

