Mathematical Reflections 2012, Issue 5 - Problem U243

Problem:
Let $f:(a,b) \longrightarrow \mathbb{R}$ be a differentiable function such that $f'(a)=f'(b)=0$ and with the
property that there is a real valued function $g$ for which $g(f'(x))=f(x)$ for all $x$ in $\mathbb{R}$.
Prove that $f$ is constant.

Proposed by Mihai Piticari and Sorin Radulescu.

Solution:
Since $f$ is differentiable in $(a,b)$ and $f'(a)=f'(b)=0$, $f$ is continuous in $[a,b]$. By Weierstrass's Theorem there exist maximum $M$ and minimum $m$ in $[a,b]$. If these are both attained at $a$ and $b$, $f$ is constant. If one of these is not attained at $a$ or $b$, then there exists $c \in (a,b)$ such that $f(c)=M$ or $f(c)=m$. Since $f$ is differentiable $f'(c)=0$, so $g(0)=f(c)=f(a)=f(b)$, which means that $f$ is constant in $(a,b)$.