# local extrema of a Continuous function

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.

• How is it 16? It's an eight degree polynomial, and can have 7 local maxima. Mar 1, 2021 at 9:08
• No the is not 6. It is 15 as the book says. Mar 1, 2021 at 10:39

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 1, 2021 at 11:24
• This is not a correct solution since f is not differentiable!!! This is not an answer. DELETE your answer immediately. Mar 2, 2021 at 8:10
• @shangq_tou: come on, you added the non-differentiability property two hours after the fact. You'd better apologize.
– user65203
Mar 2, 2021 at 8:27
• 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. Mar 2, 2021 at 8:53
• @shangq_tou: this arrogance is not necessary nor helpful, is it ?
– user65203
Mar 2, 2021 at 8:55