Question

Find the critical numbers of the function. (Enter your answers as a comma-separated list. Use n to denote any arbitrary integer values. If an answer does not exist, enter DNE.) g(θ) = 8θ − 2 tan(θ)

Answer 1

Answer:

$\theta = \cos^{-1} (\pm \frac{1}{2} )$

Step-by-step explanation:

The function is $g(\theta) = 8\cdot \theta - 2\cdot \tan \theta$, whose first derivative is:

$g'(\theta) = 8 - 2\cdot \sec^{2}\theta$

Let equalize the derivative to zero:

$8 - 2\cdot \sec^{2}\theta=0$

$\sec^{2}\theta = 4$

$1 = 4\cdot \cos^{2}\theta$

$\cos \theta = \pm\sqrt {\frac{1}{4} }$

$\cos \theta = \pm \frac{1}{2}$

The solutions are given by the following inverse trigonometric function:

$\theta = \cos^{-1} (\pm \frac{1}{2} )$