Suppose $f:[0,1]\rightarrow \mathbb{R}$ is continuous on $[0,1]$ and is differentiable on $(0,1)$. If $f(0)=0$ and $f(1)=1$ , then there exist distinct $a,b\in (0,1)$ such that $f'(a)+f'(b)=2$
I thought about Mean Value Theorem, but I only got $\exists c\in(0,1)$ such that $f'(c)=1$. However, the differentiability and continuity of f don't imply the continuity of $f'$. What's the trick here? Can someone help me? Thanks in advance.