Let $f:[0, 1] \to \mathbb{R} $ be double differentiable function such that line segment between points $(0,f(0))$ and $(1,f(1))$ pierces function at point $(a,f(a)) $ where $0<a<1$.

So now i have to prove that there exists such $x\_0$ that $f''(x\_0)=0$

So i know that the function has a second derivative, but what can i say about it, since i only have the function, not second derivative.

My first idea to somehow to use that line for describing first derivative. And i know that if i do that i will just have to show that f'(0)=f'(1) and will use Rolle's theorem to prove the statement.
Any help or tips are welcome.
Thank you in advance.