Chapter 4, Problem 19CR

(a)

To determine

The intercepts of the graph

  

Answer to Problem 19CR

Solution:

The $x$ -intercepts of the graph are $\left(-5,0\right),\left(-1,0\right),\left(5,0\right)$ and the $y$ -intercept is $\left(0,-3\right)$

Explanation of Solution

Given information:

The graph

  

  $x$ -intercepts are the points, where the graph crosses and/or touches the $x$ -axis or the point where the value of $y$ is zero and the $y$ -intercepts are the points where graph crosses the $y$ -axis or the point where the value of $x$ is zero.

By observing the graph, the graph crosses $x$ -axis at $x=-5,-1,5$ and the graph crosses $y$ -axis at $y=-3$

Therefore, the $x$ -intercepts of the graph are $\left(-5,0\right),\left(-1,0\right),\left(5,0\right)$ and the $y$ -intercept is $\left(0,-3\right)$

(b)

To determine

Whether the following graph is symmetric with respect to the $x$ -axis, the $y$ -axis and/or the origin.

The graph

  

Answer to Problem 19CR

Solution:

The graph is neither symmetric with respect to the $x$-axis, the $y$-axis nor the origin.

Explanation of Solution

Given information:

The graph

  

By observing the graph, it is observed that the graph is neither symmetric with respect to the $x$ -axis, the $y$ -axis, nor the origin.

(c)

To determine

Whether the following graph of function is a graph of even, odd, or neither function.

The graph

  

Answer to Problem 19CR

Solution:

The function is neither.

Explanation of Solution

Given information:

The graph

  

If $f\left(-x\right)=f\left(x\right)$ , then the function is an even function and if $f\left(-x\right)=-f\left(x\right)$ , then the function is an odd function, if these conditions fail, then it is neither.

Here, by observing the graph, it is observed that $f\left(-x\right)\ne f\left(x\right)$ and $f\left(-x\right)\ne -f\left(x\right)$ , so the given graph of the function is neither.

(d)

To determine

The intervals on which $f$ is increasing and decreasing, where the graph of $f$ is

  

Answer to Problem 19CR

Solution:

  $f$ is increasing on the intervals $\left(-\infty ,-3\right)\cup \left(2,\infty \right)$ and $f$ is decreasing on $\left(-3,2\right)$ .

Explanation of Solution

Given information:

The graph of $f$ is

  

By observing the graph, it is observed that $f$ is increasing on intervals $\left(-\infty ,-3\right)$ and on $\left(2,\infty \right)$ since as $x$ increases $y$ is also increasing. The function is decreasing on the interval $\left(-3,2\right)$ since as $x$ increases, $y$ also decreases.

(e)

To determine

At which, $f$ has the local maximum, if any and find that maximum value.

The graph of $f$ is,

  

Answer to Problem 19CR

Solution:

  $f$ has the local maximum at $x=-3$ and the maximum value of $f$ is $5$

Explanation of Solution

Given information:

The graph of $f$ is,

  

From the graph, it is observed that $f$ changes from increasing to decreasing at $x=-3$ . Thus, the local maximum occurs at $-3$ and the maximum value of function is $5$

Hence, the local maximum of $f$ is at $x=-3$ and the maximum value of $f$ is $5$

(f)

To determine

At which, $f$ has the local minimum, if any and find the minimum value.

The graph of $f$ is

  

Answer to Problem 19CR

Solution:

  $f$ has the local minimum at $x=2$ and the minimum value of $f$ is $-6$ .

Explanation of Solution

Given information:

The graph of $f$ is

  

From the graph, it is observed that $f$ changes from decreasing to increasing at $x=2$ . Thus, the local minimum occurs at $2$ and the minimum value of the function is $-6$

Hence, the local minimum of $f$ is at $x=2$ and the minimum value of $f$ is $-6$