Chapter 4, Problem 19CR

In parts (a) to (f), use the following graph,

Find the intercepts

Based on the graph, tell whether the graph is symmetric with respect to the x- axis, the y- axis, and/or the origin.

Based on the graph, tell whether the function is even, odd, or neither.

List the interval on which $f$ is decreasing.

List the number, if any, at which $f$ has a local maximum. What are the local maximum values?

List the number, if any, at which $f$ has a local minimum. What are the local minimum values?

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

Explanation:

$x$-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 $\left(0,-3\right)$

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

Explanation:

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.

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

Explanation:

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.

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

Explanation:

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.

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,

Explanation:

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 $5$

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

Explanation:

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 $-6$