Home > Analysis, Olympiad > IMC 2016 – Day 1 – Problem 1

## IMC 2016 – Day 1 – Problem 1

Problem 1. Let ${f:[a,b] \rightarrow \Bbb{R}}$ 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.

Solution sketch: Start from the fact that ${f}$ has infinitely many zeros in ${[a,b]}$ and pick an accumulation point of a sequence of zeros ${(x_n) \rightarrow c}$.

Now we have two cases. First we may have ${c \in \{a,b\}}$. But then ${f(c) = \lim_{n \rightarrow \infty} f(x_n)=0}$ and ${f(a)f(b) = 0}$.

Otherwise we know that ${c}$ is inside ${(a,b)}$. Then

$\displaystyle f'(c) = \lim_{x \rightarrow c} \frac{f(x)-f(c)}{x-c} = \lim_{n \rightarrow \infty} \frac{f(x_n)-f(c)}{x_n-c} = 0,$

since ${f(c) = f(x_n)= 0}$. This contradicts the hypothesis.

As an example take ${f(x) = x\sin(1/x)}$ for ${x>0}$ and ${f(0) = 0 }$.