Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a positive, smooth, periodic function with period $T$. I need to show that the infimum of $f$ is positive.
My idea is that on each interval of a period $f$ takes its minimum, which I can find by $f'(x_0)=0$ (for a suitable $x_0$). Then the infimum of $f$ is given by the number $f(x_0)$, which is positive because $f$ is assumed to be positive. But this seems to be a bit imprecise to me.
Actually, the fact that $f$ is smooth doesn't matter. All you need is that $f$ is continuous. Then the restriction of $f$ to the interval $[0,T]$ has a minimum, which is attained at some point $x_0$. But $f(x_0)>0$. And, since $f$ is periodic with period $T$,$$\min f=\min f|_{[0,T]}=f(x_0)>0.$$