Single Variable Calculus, Chapter 4, 4.1, Section 4.1, Problem 42

Determine the critical numbers of the function $g(\theta) = 4 \theta - \tan \theta$


$
\begin{equation}
\begin{aligned}

g'(\theta) =& 4 \frac{d}{d \theta} (\theta) - \frac{d}{d \theta} (\tan \theta)
\\
\\
g'(\theta) =& (4)(1) - \sec^2 \theta
\\
\\
g'(\theta) =& 4 - \sec^2 \theta

\end{aligned}
\end{equation}
$


Solving for critical numbers


$
\begin{equation}
\begin{aligned}

& g'(\theta) = 0
\\
\\
& 4 - \sec^2 \theta = 0
\\
\\
& \sec^2 \theta = 4
\\
\\
& \sqrt{(\sec \theta)^2} = \pm \sqrt{4}
\\
\\
& \sec \theta = \pm 2
\\
\\
& \frac{1}{\cos \theta} = \pm 2
\\
\\
& \cos \theta = \pm \frac{1}{2}

\end{aligned}
\end{equation}
$


Based from the unit circle, the values of $\displaystyle \cos \theta = \pm \frac{1}{2} $ are $\displaystyle \frac{\pi}{3} + 2 \pi n, \frac{5 \pi}{3} + 2 \pi n, \frac{4 \pi}{3} + 2 \pi n$ and $\displaystyle \frac{2 \pi}{3} + 2 \pi n$ (where $n$ is any integer).

Therefore, the critical numbers are $\displaystyle \frac{\pi}{3} + 2 \pi n, \frac{5 \pi}{3} + 2 \pi n, \frac{4 \pi}{3} + 2 \pi n$ and $\displaystyle \frac{2 \pi}{3} + 2 \pi n$.