Proving existence of points in a twice differentiable function

Question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function.

Prove that if $x_1<x_2<x_3$ and $f(x_1)=f(x_2)=f(x_3)=0,$ then there exists a point $q\in(x_1,x_3)$ such that $f''(q)=0$.

I think it's based on Rolle's theorem, but I can't prove it rigorously. Would anybody have a rigorous way to prove this?

Answer 1

By Rolle's theorem applied to $f$ on $[x_1, x_2]$, there is some point $y_1 \in (x_1, x_2)$ with $f'(y_1) = 0$. Similarly. $\exists y_2 \in (x_2, x_3)$ with $f'(y_2)=0$. Then apply Rolle to $f'$ on $[y_1, y_2]$, giving a point $q \in (y_1, y_2) \subset (x_1, x_3) $ with $f''(q) = 0$.