MathNet 2024-part-04 — 168 problems (EN–VI) Question 119

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

Sorin Rădulescu and Mihai Piticari