International Mathematics Competition
for University Students
2016

IMC2016: Day 1, Problem 1

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 $[0,1]$.

Proposed by Alexandr Bolbot, Novosibirsk State University

Hint: Consider an accumulation point of the zeros.

  Solution