Problem

Source: Romanian National Olympiad 2024 - Grade 11 - Problem 1

Tags: function, real analysis



Let $I \subset \mathbb{R}$ be an open interval and $f:I \to \mathbb{R}$ a twice differentiable function such that $f(x)f''(x)=0,$ for any $x \in I.$ Prove that $f''(x)=0,$ for any $x \in I.$