Today we had a probational exam in analysis. I wasn't able to solve one of the exercises and I have no idea what theorem to apply in order to solve it:
Let $I=[0,1]$ and $f: I \rightarrow \mathbb{R}$ be continuously differentiable. Assuming that $f$ has a root. Show: \max_{x \in I} |f(x)| \leq \max_{x \in I} |f'(x)|
Does this theorem have a name? What other theorem will I need in order to prove it? I'm sure the fact that the function has a root is important, but I don't see why and how to make use of it...
Thanks for your help!