Mean value theorem $f'(a)=f'(b)$

Question

Let $f$ be differentiable function such that for some $a<b$ we have $f'(a)=f'(b)$. Prove that there exists $x \in (a,b)$ such that $f(x)-f(a)=f'(x)(x-a)$.
I've already proved that without loss of generality we can assume that $f(a)=f(b)$ and I was thinking about using mean value theorem for $h(x)=\frac{f(x)-f(a)}{x-a}$, but it got me nowhere. I'd really appreciate some help with that task.