Differentiable and Continuous functions on [0.1]

Differentiable and Continuous functions on [0.1]

I've been stuck on this one for a while. Comes from an analysis qual
question.
Let f be a function that is continuous on $\left[0,1\right]$ and
differentiable on $(0,1)$. Show that if $f(0)=0$ and $|f'(x)| \leq |f(x)|$
for all $x \in (0,1)$, then $f(x)=0$ for all $x \in \left[0,1\right]$.
What I've tried doing so far is see if there was anything I could do with
MVT. I didn't really see anything to do with definitions either..to which
I have a feeling I'll be playing around with them. Drawing a picture was a
little difficult with these conditions as well
Any hints/suggestions?