Chapter SH, Problem 5.21EP

a.

To determine

Find the number of positive, negative realzeros and imaginary zeros of the given function.

Answer to Problem 5.21EP

The possible combinations of real and imaginary zeros.

Number of positive real zeros	Number of negative real zeros	Number of imaginary zeros	Total number of zeros
$3$	$1$	$0$	$4$
$1$	$1$	$2$	$4$
$0$	$0$	$4$	$4$

Explanation of Solution

Given:

The given function is $R\left(c\right)=-0.005c^4+0.25c^3+0.01c^2-2.5c+100$ .

Calculation:

Since $f\left(x\right)$ has degree $4$ , the function has $4$ zeros.

Find the type of zeros.

Examine the number of sign changes for $f\left(x\right)$ and $f\left(-x\right)$ .

  $\begin{array}{l}R\left(c\right)=-0.005c^4+0.25c^3+0.01c^2-2.5c+100\\ \text{yes}\text{no}\text{yes}\text{yes}\end{array}$

Because there are three sign changes for the coefficients of $f\left(x\right)$ , the function has three or zero positive real zeros.

  $\begin{array}{l}R\left(-c\right)=-0.005{\left(-c\right)}^4+0.25{\left(-c\right)}^3+0.01{\left(-c\right)}^2-2.5\left(-c\right)+100\\ =-0.005c^4-0.25c^3+0.01c^2+2.5c+100\\ \text{no}\text{yes}\text{no}\text{no}\end{array}$

Because there is one sign changes for the coefficients of $f\left(-x\right)$ , the function has one or zero negative real zeros.

Make a chart of the possible combinations of real and imaginary zeros.

Number of positive real zeros	Number of negative real zeros	Number of imaginary zeros	Total number of zeros
$3$	$1$	$0$	$4$
$1$	$1$	$2$	$4$
$0$	$0$	$4$	$4$

b.

To determine

Explain the meaning of the zeros in the given situation.

Explanation of Solution

Given:

The given function is $R\left(c\right)=-0.005c^4+0.25c^3+0.01c^2-2.5c+100$ .

Calculation:

Since $f\left(x\right)$ has degree $4$ , the function has $4$ zeros.

Find the type of zeros.

Examine the number of sign changes for $f\left(x\right)$ and $f\left(-x\right)$ .

  $\begin{array}{l}R\left(c\right)=-0.005c^4+0.25c^3+0.01c^2-2.5c+100\\ \text{yes}\text{no}\text{yes}\text{yes}\end{array}$

Because there are three sign changes for the coefficients of $f\left(x\right)$ , the function has three or zero positive real zeros.

  $\begin{array}{l}R\left(-c\right)=-0.005{\left(-c\right)}^4+0.25{\left(-c\right)}^3+0.01{\left(-c\right)}^2-2.5\left(-c\right)+100\\ =-0.005c^4-0.25c^3+0.01c^2+2.5c+100\\ \text{no}\text{yes}\text{no}\text{no}\end{array}$

Because there is one sign changes for the coefficients of $f\left(-x\right)$ , the function has one or zero negative real zeros.

The nonnegative zeros represents the number of calculators. The roots of the function represent the number of calculators produced per day which bring no profit to the businessman.