Chapter 4.3, Problem 32E

(a)

To determine

To find:The intervals of increase or decrease.

Explanation of Solution

Given: $f\left(t\right)=t+\cos t$

Calculate the first derivative of the given function.

  $\begin{array}{c}{f}^{\prime }\left(t\right)=1-\sin t\\ \sin t=1\\ t=-\frac{3\pi }{2},\frac{\pi }{2}\end{array}$

The function is increasing in the interval $\left(-2\pi ,\frac{-3\pi }{2}\right)$

The function is increasing in the interval $\left(-\frac{3\pi }{2},\frac{\pi }{2}\right)$

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

Therefore, the function is always increasing on $\left(-2\pi ,2\pi \right)$ .

(b)

To determine

To find :The local maxima and local minima.

Explanation of Solution

The function is always increasing on $\left(-2\pi ,2\pi \right)$ .

So, there is no local maxima and minima.

There is no local maxima and minima.

(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.

  $\begin{array}{c}{f}^{″}\left(t\right)=-\cos t\\ \cos t=0\\ t=\frac{-3\pi }{2},-\frac{\pi }{2},\frac{3\pi }{2},\frac{\pi }{2}\end{array}$

Simplify further,

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

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

  $f^{″}\left(t\right)>0$ for the interval $-\frac{3\pi }{2}<t<-\frac{\pi }{2}$ . So, the graph is concave up for $\left(-\frac{3\pi }{2},\frac{\pi }{2}\right)$

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

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

The inflection points are $\left(\frac{-3\pi }{2},-\frac{3\pi }{2}\right),\left(-\frac{\pi }{2},-\frac{\pi }{2}\right),\left(\frac{\pi }{2},\frac{\pi }{2}\right)\text{and}\left(\frac{3\pi }{2},\frac{3\pi }{2}\right)$

(d)

To determine

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

Explanation of Solution

Given: $f\left(t\right)=t+\cos t$

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

  

Figure (1)

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