Set of zeros of derivate - Lebesgue measure

Question

I'm currently struggeling with the following:
Let $\lambda$ be the Lebesgue measure and $f \in C^2[0, 1]$. Show:
If $\lambda(\{x \in [0, 1]; f(x) = 0 \}) > 0$, then $\lambda(\{ x \in [0, 1]; f'(x) = 0\}) > 0$.
Although the statement seems to be very obvious and the proof is trivial if $\{x \in [0, 1]; f(x)=0\}$ has an inner point, I don't really know how to approach the general case. I would be very grateful for a little hint.