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) = 16θ − 4 tan(θ) θ =

Answer 1

Answer:

$\theta_{1} = \frac{\pi}{3} \pm 2\pi\cdot i$, $\forall i \in \mathbb{N}_{O}$

$\theta_{2} = \frac{5\pi}{3} \pm 2\pi\cdot i$, $\forall i \in \mathbb{N}_{O}$

Step-by-step explanation:

The critical numbers are found by the First Derivative Test, which consists in differentiating the function, equalizing it to zero and solving it:

$g'(\theta) = 16 - 4\cdot \sec^{2} \theta$

Following equation needs to be solved:

$16 - 4\cdot \sec^{2}\theta = 0$

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

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

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

The solution is:

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

Given that cosine is a periodical function, there are two subsets of solution:

$\theta_{1} = \frac{\pi}{3} \pm 2\pi\cdot i$, $\forall i \in \mathbb{N}_{O}$

$\theta_{2} = \frac{5\pi}{3} \pm 2\pi\cdot i$, $\forall i \in \mathbb{N}_{O}$