Chapter 4.5, Problem 8E

a.

To determine

To calculate:The x-intercepts of the graph of the given function.

Answer to Problem 8E

The x-intercepts of the graph of the given function are $x=\left\{-\frac{3\pi }{2},-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{2},\mathrm{....}\right\}$ .

Explanation of Solution

Given information:The function is $y=f\left(x\right)=\cos x$ .

Calculation:

The function is $y=f\left(x\right)=\cos x$.........(1)

The x-intercepts are

  $\begin{array}{l}\text{Put}y=0\text{in}\text{equation}\left(\text{1}\right)\\ 0=\cos x\\ \cos x=0\\ x=-\frac{n\pi }{2},\text{where}n=-3,-1,1,3\mathrm{....}\\ x=\left\{-\frac{3\pi }{2},-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{2},\mathrm{....}\right\}\end{array}$

Hence the x-intercepts of the graph of the given function are $x=\left\{-\frac{3\pi }{2},-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{2},\mathrm{....}\right\}$ .

b.

To determine

To calculate: The y-intercept of the graph of the given function.

Answer to Problem 8E

The y-intercept of the graph of the given function is $y=1$ .

Explanation of Solution

Given information: The function is $y=f\left(x\right)=\cos x$ .

Calculation:

The function is $y=f\left(x\right)=\cos x$.........(1)

The y-interceptis

  $\begin{array}{l}\text{Put}x=0\text{in}\text{equation}\left(\text{1}\right)\\ y=\cos \left(0\right)\\ y=1\end{array}$

Hence the y-intercept of the graph of the given function is $y=1$ .

c.

To determine

To calculate: The intervals on which the graph of the given function is increasing and decreasing.

Answer to Problem 8E

The function is increasing in the intervals $x\in \left[-\pi ,0\right],\left[\pi ,2\pi \right]$ .

The function is decreasing in the intervals $x\in \left[-2\pi ,-\pi \right],\left[0,\pi \right]$ .

Explanation of Solution

Given information: The function is $y=f\left(x\right)=\cos x$ .

Calculation:

The function is $y=f\left(x\right)=\cos x$ and the graph of the function is given below.

  

As it can be observed from the graph the function is

  $\begin{array}{l}\text{decreasing}\text{for:}x\in \left[-2\pi ,-\pi \right]\\ \text{increasing}\text{for:}x\in \left[-\pi ,0\right]\\ \text{decreasing}\text{for:}x\in \left[0,\pi \right]\\ \text{increasing}\text{for:}x\in \left[\pi ,2\pi \right]\end{array}$

Hence, the function is increasing in the intervals $x\in \left[-\pi ,0\right],\left[\pi ,2\pi \right]$ .

&

The function is decreasing in the intervals $x\in \left[-2\pi ,-\pi \right],\left[0,\pi \right]$ .

d.

To determine

To calculate: The relative extrema of the graph of the given function.

Answer to Problem 8E

  $\text{The}\text{relative}\text{extrema}\text{are}\text{at}\left(x,y\right)=\left\{\left(-2\pi ,1\right),\left(-\pi ,-1\right),\left(0,1\right),\left(\pi ,-1\right),\left(2\pi ,1\right)\right\}$

Explanation of Solution

Given information: The function is $y=f\left(x\right)=\cos x$ .

Calculation:

The function is $y=f\left(x\right)=\cos x$ and the graph of the function is given below.

  

As it can be observed from the graph

  $\text{the}\text{relative}\text{extrema}\text{are}\text{at}\left(x,y\right)=\left\{\left(-2\pi ,1\right),\left(-\pi ,-1\right),\left(0,1\right),\left(\pi ,-1\right),\left(2\pi ,1\right)\right\}$

Hence $\text{the}\text{relative}\text{extrema}\text{are}\text{at}\left(x,y\right)=\left\{\left(-2\pi ,1\right),\left(-\pi ,-1\right),\left(0,1\right),\left(\pi ,-1\right),\left(2\pi ,1\right)\right\}$ .