If $f$ is a function such that $$ \int_{-\pi}^{\pi}f(x)dx=\int_c^{c+2\pi}f(x)dx \quad\forall c\in\mathbb{R}$$ and $f'(x)$exists for all $x\in\mathbb{R}$.
Then $f'$ has period $2\pi $.
I know I've to show that $f'(x)=f'(x+2\pi)$ for all $\quad x\in\mathbb{R}$. Any suggestions would be great!
By the second Fundamental Theorem of Calculus, after taking derivative for both sides w.r.t $c$, we have $$ f(c+2\pi)-f(c) =0. $$ Changing $c$ into $x$ gives $$ f(x+2\pi)=f(x). $$ Taking derivative for both sides w.r.t $x$, we have $$ f'(x+2\pi)=f'(x). $$ Namely $f'(x)$ is a periodic function with period $2\pi$.