Chapter 6.4, Problem 9CYU

To determine

To find: the consecutive value of x where the zeroes of the function is located and graph the function by collecting the domain and ranges.

Answer to Problem 9CYU

The relative maximum lied near $\text{x=-1}\text{.786}$ and that of relative minima lied near $\text{x=1}\text{.12}$ and

the zeroes of the function lied in between $\left(-3,-2\right)\cup \left(2,3\right)$ .

Explanation of Solution

Given:

  $\text{f}\left(\text{x}\right){\text{=x}}^{\text{3}}{\text{+x}}^{\text{2}}\text{-6x-3}$ .

Concept used:

The zeroes of the function resembled the function equates to zero.

  $\text{y=f}\left(\text{x}\right)\text{=0}$ .

The zero normally lies in between negative number and positive number.

Therefore, the consecutive number will negative approaches zero and positive approaches to zero except $\text{y=0}$ itself.

The graph of the function $\text{f}$ is the set of all point in the plane of the form $\left(\text{x,f}\left(\text{x}\right)\right)$ .

The graph can be defined by the graph of $\text{f}$ and the equation for the graph will be $\text{y=f}\left(\text{x}\right)$ .

The graph of the function is special case of the graph of an equation.

Relative maxima:

Which is greater point than the points directly beside it at both sides.

Whereas,

Relative minimum:

Any point which is lesser than the points directly beside it at both sides.

Calculation:

The function is given as:

  $\text{f}\left(\text{x}\right){\text{=x}}^{\text{3}}{\text{+x}}^{\text{2}}\text{-6x-3}$ .

The table for the graph of the function will be:

By putting the value $\text{x}$ the value of $\text{y}$ can be determine.

The domain here is $\text{x}$ and range here is $\text{y}$ .

$\text{x}$	$-3$	$-2$	$\text{-1}$	$\text{0}$	$\text{1}$	$\text{2}$	$3$
$\text{f}\left(\text{x}\right)$	$-3$	$5$	$3$	$-3$	$-7$	$-3$	$15$

The points in the graph so collected will form a graph for the given equation.

The changes in sign from the result indicates that there are zeroes between $\text{x=-3}$ and $\text{x=-2}$ ,and between $\text{x=2}$ and $\text{x=3}$ .

Since, the sign of the function changes from negative to positive.

The graph of the given equation $\text{f}\left(\text{x}\right){\text{=-3x}}^{\text{4}}{\text{+5x}}^{\text{3}}{\text{+4x}}^{\text{2}}\text{+4x-8}$ by collecting the points will be:

  

The relative maxima and minima can be calculated as:

The value of $\text{f}\left(\text{x}\right)$ at $\text{x=-1}\text{.786}$ greater than the surrounding points or the curve is open downward so there must be relative maximum near $\text{x=-1}\text{.786}$

The value of $\text{f}\left(\text{x}\right)$ at $\text{x=1}\text{.12}$ is less than the surrounding points or the curve is open upward so there must be relative minimum near $\text{x=1}\text{.12}$ .

Hence relative maximum lied near $\text{x=-1}\text{.786}$ and that of relative minima lied near $\text{x=1}\text{.12}$ and

the zeroes of the function lied in between $\left(-3,-2\right)\cup \left(2,3\right)$ .