Chapter 2.3, Problem 10E

Solution

To Match: The given polynomial function with its graph.

For the polynomial function $f\left(x\right)=-9x^3+27x^2+54x-73$ the matching graph is option $\left(b\right)$ .

Given: $f\left(x\right)=-9x^3+27x^2+54x-73$

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 $f\left(x\right)=-9x^3+27x^2+54x-73$ is a cubic function 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 first derivative of the function $f\left(x\right)=-9x^3+27x^2+54x-73$ :

  $f\left(x\right)=-27x^2+54x+54$

Find the second derivative of the function:

  $f^{\prime \prime }\left(x\right)=-54x+54$

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

  $\begin{array}{c}-27x^2+54x+54=0\\ -27\left(x^2-2x-2\right)=0\\ \left(x^2-2x-2\right)=0\end{array}$

Use the quadratic formula to find the solutions:

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

Substitute the values $a=1,b=-2$ , and $c=-2$ into the quadratic formula and solve for $x$ :

  $\frac{2\pm \sqrt{{\left(-2\right)}^2-4\cdot \left(1\cdot \left(-2\right)\right)}}{2\cdot 1}$

Simplify the equation:

  $x=1\pm \sqrt{3}$

The local maximum and local minimum are:

  $\begin{array}{c}x=1+\sqrt{3}\text{or}\\ x\approx 2.732\\ x=1-\sqrt{3}\text{or}\\ x\approx -0.732\end{array}$

The function $f$ has a local maximum of about $92.53$ at $x\approx 2.732$ , and a local minimum of about $-94.53$ at $x\approx -0.732$ .

Find the zeros at $x-$intercept:

Graph each side of the equation $-9x^3+27x^2+54x-73=0$ . The solution is the $x-$ value of the point of intersection.

  $-2.00615389$ (Multiplicity of 1)

  $1.01234588$ (Multiplicity of 1)

  $3+\frac{4\sqrt{5}}{9}$ (Multiplicity of 1)

  

The function has zeros at $-9x^3+27x^2+54x-73=0$ . Therefore, the graph is option $\left(b\right)$ .

  

Thus, for the polynomial function $f\left(x\right)=7x^3-21x^2-91x+104$ the matching graph is option (b).