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Expert-verified Found in: Page 209 ### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280 # Find the critical numbers of the function.34. $$g(\theta ) = 4\theta - \tan \theta$$.

The critical numbers of the function $$g(\theta ) = 4\theta - \tan \theta$$ are $$\theta = 2n\pi \pm \frac{\pi }{3},\theta = 2n\pi \pm \frac{{2\pi }}{3}$$, for all $$n = 0,1,2,3 \ldots ..$$

See the step by step solution

## Step 1: Given data

The function is $$g(\theta ) = 4\theta - \tan \theta$$.

## Step 2: Concept of critical number

A critical number of a function $$f$$ is a number $$c$$, if it satisfies either of the below conditions:

(1) $${f^\prime }(c) = 0$$

(2) $${f^\prime }(c)$$, Does not exist.

## Step 3: Obtain the first derivative of the given function

Obtain the first derivative of the given function.

\begin{aligned}{c}{g^\prime }(\theta ) &= \frac{d}{{d\theta }}(4\theta - \tan \theta )\\ &= \frac{d}{{d\theta }}(4\theta ) - \frac{d}{{dt}}(\tan \theta )\\ &= 4 - {\sec ^2}\theta \end{aligned}

Take $${g^\prime }(\theta ) = 0$$and obtain the critical numbers.

\begin{aligned}{c}4 - {\sec ^2}\theta &= 0\\{\sec ^2}\theta &= 4\\\sec \theta &= \pm 2\\\cos \theta &= \pm \frac{1}{2}\end{aligned}

Therefore, $$\cos \theta = \frac{1}{2}$$ and $$\cos \theta = \frac{{ - 1}}{2}$$.

## Step 4: Obtain the critical numbers of the function $$g(\theta ) = 4\theta - \tan \theta$$

Compute the value of $$\theta$$ for the two possible cases.

Recall the fact that the general solution of the equation $$\cos \theta = \cos \beta$$:

$$\theta = 2n\pi \pm \beta$$. ……. (1)

Case (1):

Consider, $$\cos \theta = \frac{1}{2}$$. ……. (2)

Express equation (2).

$$\cos \theta = \cos \left( {\frac{\pi }{3}} \right)$$ ……. (3)

Apply the general solution as shown in equation (1) and obtain the solution for equation (3).

$$\theta = 2n\pi \pm \frac{\pi }{3}$$

Therefore, the general solution of equation (3) is $$\theta = 2n\pi \pm \frac{\pi }{3}$$.

Case (2):

Consider, $$\cos \theta = \frac{{ - 1}}{2}$$. ……. (4)

Express the equation (3).

$$\cos \theta = \cos \left( {\frac{{2\pi }}{3}} \right)$$ ……. (5)

Apply the general solution as shown in equation (1) and obtain the solution for equation (5).

$$\theta = 2n\pi \pm \frac{{2\pi }}{3}$$.

The critical numbers of the function $$g(\theta ) = 4\theta - \tan \theta$$ are $$\theta = 2n\pi \pm \frac{\pi }{3},\;\theta = 2n\pi \pm \frac{{2\pi }}{3}$$, for all $$n = 0,1,2,3 \ldots ..$$ ### Want to see more solutions like these? 