No worries, you need only determine when $f(x)$ is increasing or decreasing on for $x \in [0, 2\pi]$
Test between $x = 0$ and $x \lt \pi/3$ (the derivative is negative there).
Test between $x > \pi/3$ and $x < 5\pi/3$, (the derivative is positive there) and then
test the values between $x > 5\pi/3$ and $x\leq 2\pi$ (where the $f'(x)\lt 0$).
The other "zeros" you see occur outside of the interval $x \in [0, 2\pi]$.
So, you need only worry $f(x)$ on the interval $[0, 2\pi]$.
$f(x)$ is decreasing on the intervals $x\in [0, \pi/3)$ and increases on $x\in (\pi/3, 5\pi/3)$, then decreases on the interval $x \in (5\pi/3, 2\pi]$.