I'm trying to solve this problem
Let $f:[a,b] \to \mathbb R$ a differentiable function with continuous derivative.
Suppose further that $f$ is twice differentiable on $(a,b)$.
Prove that if $f(a)=f(b)$ and f'(a)=f'(b)=0, then exist $x_1, x_2 \in (a,b)$ with $x_1 \neq x_2$ such that f''(x_1) = f''(x_2).
I'm trying to solve this problem graphically intuitively ; I tried, and likely the problem is real: if $f(a) = f(b)$ and f'(a) = f'(b) = 0, the points $a$ and $b$ are the points of maximum and minimum for the function. but how can you prove that there are $x_1, x_2 \in (a,b)$ with $x_1 \neq x_2$ such that f''(x_1) = f''(x_2)?