Step 3: Determine the values of $x$ for which $f$ is increasing. The function is $f(x) = 2\cos x + 1$. To find where $f(x)$ is increasing, we need to find its derivative $f'(x)$ and determine where $f'(x) > 0$. $$ f'(x) = \frac{d}{dx}(2\cos x + 1) = -2\sin x $$ We need $f'(x) > 0$, so $-2\sin x > 0$. Dividing by $-2$ and reversing the inequality sign gives $\sin x < 0$. In the interval $x \in [-180^\circ, 1