For what $f$ is $\int_0^{2\pi}f(x)\cos(2nx)dx\geq 0 ~~\forall n\in\mathbb Z$

Question

I have $f$ is positive, even, $2\pi$ periodic and continuous function on $\mathbb R$. Let $a_n=\int_0^{2\pi}f(x)\cos(2nx)dx.$

Is $a_n\geq 0$ for all $n\in\mathbb Z$ ?

I tried in many ways, but not able to prove or disprove.

Please help me!

Answer 1

The answer is no.

Lets consider $f(x)= π^3-(x - \pi)^2 $ on $[0,2\pi]$ we have that $$ a_1=\int_0^{2π}( π^3-(x - π)^2) cos(2x) dx = -\pi<0 $$.