Let $f$ be an Infinitely differentiable function on $[0,1]$ and suppose for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{n}(x)=0$. Then does $f$ coincide on $[0,1]$ with some polynomial? If yes then how.

I though of using Weierstrass Approximation theorem, but couldn't succeed.