Let $f$ be a continuously differentiable function on $[a,b]$ with $f(a)=f(b)$ and $f'(a)=f'(b)$. Then, do there exist $x_1,x_2\in(a,b)$ such that $f'(x_1)=f'(x_2)$ for $x_1\neq x_2$?
I think the answer is yes by intermediate value theorem, but am not getting the sufficient rigour to prove it. Any hints. Thanks beforehand.