IMC 2016 · Problem 1

Let $f \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$. Suppose that $f$ has infinitely many zeros, but there is no $x \in (a, b)$ with $f(x) = f'(x) = 0$.

(a) Prove that $f(a)f(b) = 0$.

(b) Give an example of such a function on $[0, 1]$.