Let $f: [0,1]\rightarrow R$ be a continuous function such that $f(0)=0$ and it's differentiable on $(0,1)$. In addition, we know that $0\le f'(x)\le2f(x)$. Prove that $f=0$.
I know that if the function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$ and $f'(x) = 0$ on $(a,b)$, then $f$ must be a constant function on $[a,b]$.
But I do not quite know how to use the inequality.