Let $g:\mathbf{R\rightarrow R}$ be a twice differentiable function.
We are supposed to prove that:
If $a_1 < a_2 < a_3$ and $g(a_1)=g(a_2)=g(a_3)=0$, prove that there exists a point $q \in (a_1,a_3)$ such that $g''(q)=0$.
Does anyone know how this can be done?
Rolle’s theorem gives $g’(c_1)=0$ and $g’(c_2)=0$ for at least one $c_1\in(a_1,a_2)$ and a $c_2\in(a_2,a_3)$. As $g’(c_1)=g’(c_2)$, applying Rolle’s to $g’(x)$ gives $g’’(c)=0$ for $c\in (c_1,c_2)\subset(a_1,a_3)$.