Chapter 2.3, Problem 11E

Solution

To Match: The given polynomial function with its graph.

a

For the polynomial function $f\left(x\right)=x^5-8x^4+9x^3+58x^2-164x+69$ the matching graph is option $\left(a\right)$ .

Given: $f\left(x\right)=x^5-8x^4+9x^3+58x^2-164x+69$

Calculation:

For any polynomial function $f\left(x\right)=a_n{x}^n+\dots +a_{1}x+a_0$ , the limits $\underset{x\to \infty }{\lim }f\left(x\right)$ and $\underset{x\to -\infty }{\lim }f\left(x\right)$ are determined by the degree $n$ of the polynomial and its leading coefficients $a_n$ .

Here the function $f\left(x\right)=x^5-8x^4+9x^3+58x^2-164x+69$ is a polynomial of degree $5$ and the leading coefficients is $a_n>0.$

So, the limits are $\underset{x\to -\infty }{\lim }f\left(x\right)=-\infty$ and $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$ .

Find the local maximum and local minimum:

Find the derivative of the term $f\left(x\right)=x^5-8x^4+9x^3+58x^2-164x+69$ :

  $5x^4-32x^3+27x^2+116x-164$

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve:

  $\begin{array}{c}5x^4-32x^3+27x^2+116x-164=0\\ \left(x-2\right)\left(5x^3-22x^2-17x+82\right)=0\\ \left(x-2\right)\left(x-2\right)\left(5x^2-12x-41\right)=0\\ {\left(x-2\right)}^2\left(5x^2-12x-41\right)=0\end{array}$

Solve for $x:$

Set ${\left(x-2\right)}^2$ equal to $0$ :

  $\begin{array}{c}{\left(x-2\right)}^2=0\\ x=2\end{array}$

Set $\left(5x^2-12x-41\right)$ equal to $0$:

Use the quadratic formula to find the solutions.

  $\frac{-b\pm \sqrt{b^2-4\left(ac\right)}}{2a}$

Substitute the values $a=5,b=-12,$ and $c=-41$ into the quadratic formula and solve for $x.$

  $\frac{12\pm \sqrt{{\left(-12\right)}^2-4\cdot \left(5\cdot -41\right)}}{2\cdot 5}$

Simplify the term:

  $x=\frac{6\pm \sqrt{241}}{5}$

The $x-$ values are:

  $\begin{array}{c}x=2,\\ x=\frac{6+\sqrt{241}}{5}\text{or}\\ x=0.521,\\ x=\frac{6-\sqrt{241}}{5}\text{or}\\ x=\mathrm{5.191.}\end{array}$ .

  

The function $f$ has a local maximum of about $399.236$ at $x\approx -1.904$ , and a local minimum of about $-113.175$ at $x\approx 4.304$ .

Find the zeros at $x-$intercept:

The function has zeros at $x=-2.946,x=0.521,x=5.191$ .

The graph $b$ and $d$ does not have $\underset{x\to -\infty }{\lim }f\left(x\right)=-\infty$ and $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$ and the graph $c$ is a cubic function as it has three roots. Therefore, the graph is option $\left(a\right)$

Thus, for the polynomial function $f\left(x\right)=x^5-8x^4+9x^3+58x^2-164x+69$ the matching graph is option $\left(a\right)$ .