Chapter 4.3, Problem 31E

(a)

To determine

To find:The intervals of increase or decrease.

Explanation of Solution

Given: $f\left(\theta \right)=2\cos \theta +{\cos }^2\theta$

Calculate the first derivative of the given function.

  $\begin{array}{c}{f}^{\prime }\left(\theta \right)=-2\sin \theta +2\cos \theta \left(-\sin \theta \right)\\ =-2\sin \theta \left(\cos \theta +1\right)\end{array}$

The function is increasing in the interval $\left(\pi ,2\pi \right)$

The function is decreasing in the interval $\left(0,\pi \right)$ .

(b)

To determine

To find :The local maxima and local minima.

Explanation of Solution

The function is increasing in the interval $\left(\pi ,2\pi \right)$

The function is decreasing in the interval $\left(0,\pi \right)$ .

The value of $f^{\prime }$ is changing from negative to positive at $\theta =\pi$

So, the local minima is $\left(\pi ,-1\right)$

There is no local maxima.

(c)

To determine

To find :The concavity of the function and the point of inflection.

Explanation of Solution

Calculate the second derivative of the given function.

  $\begin{array}{c}{f}^{″}\left(\theta \right)=-2\cos \theta \left(\cos \theta +1\right)-2\sin \theta \left(-\sin \theta \right)\\ =-2{\cos }^2\theta -2\cos \theta +2{\sin }^2\theta \\ =-2\left(\cos 2\theta +\cos \theta \right)\\ 0=\cos 2\theta +\cos \theta \end{array}$

Simplify further,

  $\begin{array}{c}2\cos \left(\frac{2\theta +\theta }{2}\right)\cos \left(\frac{2\theta -\theta }{2}\right)=0\\ 2\cos \left(\frac{3\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)=0\\ \theta =\frac{\pi }{3},\frac{5\pi }{3},\pi \end{array}$

  $f^{″}\left(\theta \right)>0$ for the interval $0<\theta <\frac{\pi }{3}$ . So, the graph is concave up for $\left(0,\frac{\pi }{3}\right)$

  $f^{″}\left(x\right)<0$ for the interval $\frac{\pi }{3}<\theta <\pi$ . So, the graph is concave down for $\left(\frac{\pi }{3},\pi \right)$

  $f^{″}\left(\theta \right)>0$ for the interval $\pi <\theta <\frac{5\pi }{3}$ . So, the graph is concave up for $\left(\pi ,\frac{5\pi }{3}\right)$

  $f^{″}\left(\theta \right)<0$ for the interval $\frac{5\pi }{3}<\theta <2\pi$ . So, the graph is concave down for $\left(\frac{5\pi }{3},2\pi \right)$

The inflection points are $\left(\frac{\pi }{3},\frac{5}{4}\right),\left(\frac{5\pi }{3},\frac{5}{4}\right)\text{and}\left(\pi ,-1\right)$

(d)

To determine

To sketch :The graph of result obtained from part (a) to (c).

Explanation of Solution

Given: $f\left(\theta \right)=2\cos \theta +{\cos }^2\theta$

The graph of the result obtained is shown in figure below.

  

Figure (1)

Therefore, the required graph is shown in figure (1).