Let $f$ a periodic function which is differentiable on $\mathbb R$

Question

Let $f$ a periodic function which is differentiable on $\mathbb R$

1)prove that the equation $f^\prime(x)=0$ has may solutions on $\mathbb R$

2)Prove that: $f$ is a bounded function

I cant start the problem because i have no idea to how we can solve it

Answer 1

Hint If $f(a)=f(b)$ then by Rolle's Theorem there is a root between $a$ and $b$.

Hint 2 $f(x)=f(x+T)=f(x+2T)=....$

Hint 3 $f$ is bounded on $[0,T]$ (Why?)

Answer 2

Hints:

(2) Suppose $\;T>0\;$ is the function's period, then $\;f(\Bbb R)=f([0,T])\;$, and since $\;f\;$ is continuous, with Weierstrass we're done

(1) Since $\;f(1)=f(1+T)\;$, rolle's theorem tells us that at some point $\;f'(c)=0\;$ .