Chapter 2.2, Problem 93E

(a)

To determine

To find: The intermediate value theorem and the table feature of the graphing utility to find the intervals one unit in length in which the polynomial function is guaranteed to have zero.

Answer to Problem 93E

Theby the intermediate value theorem and by the table features it is concluded that the unit length intervals are $\left[-1,0\right],\left[1,2\right]$ and $\left[2,3\right]$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=x^3-3x^2+3$

Calculation:

Consider the value of the function for the different value of the $x$ is shown in Table 1

Table 1

$x$	$-1$	$0$	$1$	$2$	$3$
$f\left(x\right)$	$-1$	$3$	$1$	$-1$	$3$

For $f\left(-1\right)$ is negative and $f\left(0\right)$ is positive, the intermediate value theorem is applied. As the function has zero between $-1$ and $0$ , In the same way it can be concluded that that function will have zero between the 1 and 2 and between 2 and $3$ .

(b)

To determine

To find: The graph for the function to verify the zeros of the function.

Explanation of Solution

The graph for the polynomial to find the exact zero location. Such that the graph of the polynomial function is shown below in Figure 1

  

Figure 1

From the graph it is clear that the zeros lies between the points $-0.879,1.347$ .