RO-NMO 2008 Problem 113

Let $f: \mathbb R \to \mathbb R$ be a function, two times derivable on $\mathbb R$ for which there exist $c\in\mathbb R$ such that $$\frac { f(b)-f(a) }{b-a} \neq f'(c) ,$$ for all $a\neq b \in \mathbb R$ .

Prove that $f''(c)=0$ .