Assume that $f$ continuous at $[a,b]$ and differentiable at $(a,b)$. Prove that if $f''(x) \neq 0$ then $f$ gets every value at $[a,b]$ twice,at most

Question

Assume that $f$ is continuous at $[a,b]$ and differentiable at $(a,b)$. Prove that if $f''(x) \neq 0$ for every $x \in (a,b)$ then $f$ gets every value at $[a,b]$ twice, at most.

I tried to prove it by assuming that there exist $x_1 > x_2 > x_3 $ such that $f(x_1) = f(x_2) = f(x_3)$. but can not really understand how to use the assumption that $f''(x) \neq 0$ for every $x \in (a,b)$

I suppose you need to use Rolle's theorem but I didn't get there just yet.

Answer 1

Let's use your assumption with $x1 > x2 > x3$.

Because $f$ is continuous over $[x2, x1]$,differentiable over $]x2, x1[$ and $f(x_2)=f(x_1)$,

$$ \exists c_1 \in ]x2, x1[, f'(c_1) = 0$$

The same way,

$$ \exists c_2 \in ]x3, x2[, f'(c_2) = 0$$

Do you think it will be possible to apply Rolle's theorem on $f'$ over $[c2, c1]$?