Chapter 6.4, Problem 12CYU

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 12CYU

There is no relative maximum and minimum the zeroes of the function lied in between $\left(1,2\right)$ .

Explanation of Solution

Given:

  $\text{f}\left(\text{x}\right){\text{=-x}}^{\text{3}}{\text{+2x}}^{\text{2}}\text{-3x+4}$ .

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:

  $\text{f}\left(\text{x}\right){\text{=-x}}^{\text{3}}{\text{+2x}}^{\text{2}}\text{-3x+4}$ .

The table for the graph of the function will be:

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

$\text{x}$	$-4$	$-3$	$-2$	$\text{-1}$	$\text{0}$	$\text{1}$	$\text{2}$	$3$	$4$
$\text{f}\left(\text{x}\right)$	$112$	$58$	$26$	$10$	$4$	$2$	$-2$	$-14$	$-40$

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 $\text{x=1}$ 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:

  

There is no relative maximum and minimum since there is no point which can make angle $0^0$ with the x-axis which means the tangent line will make parallel to the x-axis. Therefore, there is no such inflection or extreme point where function changes its slope.

Hence, there is no relative maximum and minimum the zeroes of the function lied in between $\left(1,2\right)$ .