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Let be a continuous function $f\colon \mathbb{R}\to \mathbb{R}\:$ that has exact 3 local extrema. $f$ is NOT differentiable. Find the maximum number of local extrema that the function $f\circ f$ can have.

I do not know how to prove that but I obtained from my intuition that it must be 15. If we take for example a 8-degree polynomyal then $f\circ f$ is a 16-degree polynomyal and it does have maximum 15 local extrema.

The options of this multiple choice exercise are:

A)10 B)3 C)15 D)16

And the book says that the correct answer is C)15. And I don't know why. I need a complete proof.

shangq_tou
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1 Answers1

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By the chain rule,

$$(f(f(x))'=f'(f(x))f'(x)$$ where $f'$ denotes the formal derivative of $f$.

We know that $f'$ cancels exactly $3$ times, and that $f(x)$ can repeat the same value $4$ times. Hence the derivative can have

$$4\cdot3+3$$ roots and the function

$$15$$ extrema.

  • $f$ is not given to be differentiable. – Koro Mar 01 '21 at 11:24
  • This is not a correct solution since f is not differentiable!!! This is not an answer. DELETE your answer immediately. – shangq_tou Mar 02 '21 at 08:10
  • @shangq_tou: come on, you added the non-differentiability property two hours after the fact. You'd better apologize. –  Mar 02 '21 at 08:27
  • If f is continuous than it doesn't necessarily mean that f is differentiable. But it seems like you didn't learn the theory. I added that f is differentiable because of the fact that not many people like you seem to know that fact, unfortunately. – shangq_tou Mar 02 '21 at 08:53
  • @shangq_tou: this arrogance is not necessary nor helpful, is it ? –  Mar 02 '21 at 08:55