#### Answer

The critical numbers are $\pi n$

#### Work Step by Step

Find the critical numbers of $f(\theta ) = 2\cos \theta + \sin ^2 \theta$
Differentiate and set the derivative $=0$
$f'(\theta) = -2\sin \theta + 2\sin \theta \cos \theta$
$0=2\sin \theta (\cos \theta - 1)$
$f'(\theta) = 0$ when $\sin \theta = 0$ and $\cos \theta = 1$
Therefore, the critical numbers are $\pi n$ and $2\pi n$, which can be simplified down to $\pi n$.