I came across the following exercise in Spivak's book: Suppose that you have a function $f$ such that $f'(x) \ge M > 0$, for any $x \in [0,1]$. I would like to show that there is an interval of length $1/4$ contained in $[0,1]$, on which $|f(x)| \ge M/4$.
My feeling is that this can be shown using the Mean Value Theorem: Applying this yields, for any $x \in (0,1]$, that $ \frac{f(x) - f(0)}{x-0} \ge M. $ However, I don't see how to proceed from here, as I don't have a lower bound for $f(0)$.