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 }.

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