IMC 2016 Problem 1

Let $f : \left[ a, b\right]\rightarrow\mathbb{R}$ be continuous on $\left[ a, b\right]$ and differentiable on $\left( a, b\right)$ . Suppose that $f$ has infinitely many zeros, but there is no $x\in \left( a, b\right)$ with $f(x)=f'(x)=0$ . (a) Prove that $f(a)f(b)=0$ . (b) Give an example of such a function on $\left[ 0, 1\right]$ .

(Proposed by Alexandr Bolbot, Novosibirsk State University)