Let $f$ be continuously differentiable on $[a,b]$, and there exists a $c\in (a,b)$ such that $f'(c)=0$. Prove that there exists $\eta\in (a,b)$ such that $$f'(\eta)=\frac{f(\eta)-f(a)}{b-a}.$$

I could only see that $f$ attains its minimum or maximum in $(a,b)$.