i'd like your thoughts on this one. thank you!

Find the intervals on which f is increasing/decreasing and state any relative extremaSo I find the derivative to find the critical numbers:1)$\displaystyle f(\theta) = 2cos\theta - cos2\theta\,,\,[0,2\pi]$

$\displaystyle f\,'\,(\theta) = 2(\,-\,sin\theta) - (\,-\,sin\,2\theta)(2)$

$\displaystyle = -2sin\theta + 2sin\,2\theta$

So I set that equal to zero: $\displaystyle -2sin\,\theta + 2sin\,2\theta = 0$

Divide by -2: $\displaystyle sin\theta - sin2\theta = 0$

Add $\displaystyle sin 2\theta$: $\displaystyle sin\theta = sin 2\theta$

Sub $\displaystyle sin2\theta = 2sin\theta cos\theta$: $\displaystyle sin\theta = 2sin\theta cos\theta$

Divide by $\displaystyle sin \theta$: $\displaystyle 1 = 2cos\theta$

So: $\displaystyle cos\theta = \frac{1}{2}$

So I went on the unit circle and found all the places where the x coordinate is one half and included the negative of those angles angles since $\displaystyle cos (\,-\,x) = cosx$

Based on that I got $\displaystyle \theta = \pm\frac{\pi}{3}\,,\,\pm\frac{5\pi}{3}$

Is this right so far?