Chapter 11, Problem 12CR

(a)

To determine

To graph: The function $f\left(x\right)=x^3-3x+5$ using a graphing utility, and approximate the zero(s) of $f$.

Explanation of Solution

Given information:

The function $f\left(x\right)=x^3-3x+5$.

Graph:

Let $y=x^3-3x+5$

Steps for graphing an equation using a graphing utility:

Step 1: Press ‘ON’ key.

Step 2: Press [Y=] key. From the $Y_1=$ screen, enter the expression after prompt $x^3-3x+5$.

Step 3: Press $[\text{GRAPH}]$ key.

The window shows the graph of the function $f\left(x\right)=x^3-3x+5$ as,

It is observed that, the zero of $f\left(x\right)=x^3-3x+5$ is approximately $x=-2.279$.

Interpretation:

The graph shows the graph of a function $f\left(x\right)=x^3-3x+5$. The approximate zero of $f\left(x\right)=x^3-3x+5$ is $-2.279$.

(b)

To determine

The local maxima and local minima using graphing utility.

Answer to Problem 12CR

Solution:

The local maxima is $7$ and local minima is $3$.

Explanation of Solution

Given information:

The function is $f\left(x\right)=x^3-3x+5$.

Explanation:

A local maximum point on a function is a point $\left(x,y\right)$ on the graph of the function whose $y$ coordinate is larger than all other $y$ coordinates on the graph at points close to $\left(x,y\right)$.

A local minimum point on a function is a point $\left(x,y\right)$ on the graph of the function whose $y$ coordinate is smaller than all other $y$ coordinates on the graph at points close to $\left(x,y\right)$.

From the part (a), the graph of the function is

By looking the it is observed that local maxima is $7$ which is occur at $x=-1$ and local minima is $3$ which is occur at $x=1$.

(C)

To determine

The interval on which $f$ is increasing.

Answer to Problem 12CR

Solution:

The increasing interval of a function is $\left(-\infty ,-1\right)\cup \left(1,\infty \right)$.

Explanation of Solution

Given information:

The function is $f\left(x\right)=x^3-3x+5$.

Explanation:

A function is increasing when the $y-$ value increases as the $x-$ value increases.

From the part (a), the graph of the function is

 By looking the graph, the function is increasing for the interval $\left(-\infty ,-1\right)$ and $\left(1,\infty \right)$.

Therefore, increasing interval of a function is $\left(-\infty ,-1\right)\cup \left(1,\infty \right)$.