Chapter 4.2, Problem 85AYU

In Problems 91-98, analyze each polynomial function using Steps 1 through 8 on page 193 in Section 4.1.

f ( x )   =   6 x 4   −   37 x 3   +   58 x 2   +   3 x   −   18

To determine

To analyze: The polynomial function $6x^4-37x^3+58x^2+3x-18$.

Answer to Problem 85AYU

$\text{The polynomial function }f\left(x\right)=6x^4-37x^3+58x^2+3x-18\text{ is analyzed}$.

Explanation of Solution

Given:

$6x^4-37x^3+58x^2+3x-18$

Step 1: Determine the end behavior of the function

The polynomial function $f$ is of degree 4. The graph of $f$ behaves like $y=x^4$ for large values of $\left|x\right|$.

Step 2: Finding the $x$ and $y\text{-intercepts}$ of the graph of the function

To find the $y\text{-intercept}$, finding $f\left(0\right)$

$6x^4-37x^3+58x^2+3x-18$

$f\left(0\right)=-18$

$y\text{-intercept }=-18$

To find the $x\text{-intercept}$, solving $f\left(x\right)=6x^4-37x^3+58x^2+3x-18=\text{ 0}$

Using synthetic division

$\left(x-3\right)\left(6x^3-19x^2+x+6\right)=\left(x+3\right)\left(2x+1\right)\left(3x-2\right)\left(x-3\right)$

$x\text{-intercept }=\text{ 3, }-3,-\frac{1}{2},+\frac{2}{3}$

Step 3: Determining the zeros of the function and their multiplicity to find the graph crosses or touches the $x\text{-axis}$ at each intercept.

The zeros 3, $-3$, $-\frac{1}{2}$, $+\frac{2}{3}$ each have multiplicity 1. Therefore at $-3$, $\frac{1}{\sqrt{2}}$, $-\frac{1}{\sqrt{2}}$ the graph crosses the $x\text{-axis}$

Step 4: Using the graphing utility to graph the function.

Step 5: Approximate the turning point of the graph

From the graph of $f$, we see $f$ has 3 turning points

Using the graph maximum turning point $=\left(1.7,\text{ 23.5}\right)$ rounded to two decimal places

Using the graph minimum turning point $=\left(0,-18\right)$ rounded to two decimal places

Step 6: Redraw the graph

The graph passes through $\left(-3,\text{ 0}\right)$ and $\left(-0.50,0\right)$

The end behavior of the graph $=y=x^4$

Step 7: Find the domain and range of the function

The domain and range is the set of all real numbers

Step 8: Use the graph to determine where the function is increasing or where it is decreasing

Based on the graph, $f$ is increasing on the intervals $\left(-18,0.8\right]$ and $\left[3,\infty \right)$

Based on the graph, $f$ is decreasing $\left(\infty ,-18\right]\left[0.8,3\right]$