Let $f(x)$ be a real valued function whose first, second derivatives are continuous in $[0,2\pi]$. Also, $f''(x)\geq0$.
Show that $\int_{0}^{2\pi}f(x)\cos{x} dx\geq 0$.
I tried using Integration by parts twice, by which I got
$I= \int_{0}^{2\pi}f(x)\cos{x} dx\geq 0$
$\implies f'(2\pi)-f'(0)\geq\int_{0}^{2\pi}f''(x)\cos{x}dx$
I know that this LHS should be $\geq0$, but how do I compare it with RHS?
Any hints, solutions or suggestions would be really helpful. Thank you!
