Let $f$ be a function which is differentiable on $[0, \frac{\pi}{2}]$, such that $0\leq f'(x)\leq1$ for all $x$ in this interval. I'm asked to prove that there exists $x\in [0, \frac{\pi}{2}]$ such that $f'(x)=\sin x$.
I believe that I should use the Darboux's theorem, but I fail to do it here.