Chapter 6.4, Problem 43PPS

(a)

To determine

To graph: For the given function.

Explanation of Solution

Given information:

The given equation is $y=x^5-3x^3+2x-4$

Graph:

Using MAPLE to graph the given function using the ranges as shown

  

Figure (1)

Interpretation: Graph for the function

  $f\left(x\right)=x^5-3x^3+2x-4$ is shown in figure (1).

The number of zeroes is the number of intersection made on x- axis. From the above graph the number of intersection made is two

Therefore, the number of zeroes is $1$ And the zero is $x=175$ .

  

Figure 2

Therefore, the x-intercept is $\text{1}.\text{75},$ and the y-intercept is $-4$ .

The value of $f\left(x\right)$ at $x\approx -\text{1}.\text{25}$ and at is $x\approx -0.\text{5}$ greater than the surrounding point

So

  $x=-\text{1}.\text{25},-0.\text{5},0.\text{5},\text{1}.\text{25}$

  $x\approx -\text{1}.\text{25},0.\text{5}$ are turning points

The value of $f\left(x\right)$ at $x\approx -0.\text{5}$ and at $x\approx -\text{1}.\text{25}$ is less than the surrounding points

So $x\approx -0.\text{5},\text{1}.\text{25}$ ,are turning points.

Therefore, the points are mad on $x=-\text{1}.\text{25},-0.\text{5},0.\text{5},\text{1}.\text{25}$ .

(b)

To determine

To find: The axis of symmetry.

Answer to Problem 43PPS

There is no line of symmetry.

Explanation of Solution

Given information:

The given equation is $f\left(x\right)=x^5-3x^2+2x-4$

Calculation:

Line symmetry is the line which divides the given graph into two equal halves

The function does not have an axis of symmetry because the graph is parabola.

Therefore, there is no line of symmetry.

(c)

To determine

To find: The intervals for which it is increasing, decreasing, or constant

Answer to Problem 43PPS

The graph is increasing in the interval $x\le -1.25$ , $-0.\text{5}\le \text{x}\le 0.\text{5}$ and $\text{x}\ge \text{1}.\text{25}$ and decreasing in $-\text{1}.\text{25}\le \text{x}\le -0.\text{5}$ and $0.\text{5}\le \text{x}\le \text{1}.\text{25}$

Explanation of Solution

Given information:

The given equation is $f\left(x\right)=x^4-8x^2+16$

Calculation:

The curve is increasing when the curve going upwards and is decreasing when the curve is going downwards.

Therefore, The graph is increasing in the interval $x\le -1.25$ , $-0.\text{5}\le \text{x}\le 0.\text{5}$ and $\text{x}\ge \text{1}.\text{25}$ and decreasing in $-\text{1}.\text{25}\le \text{x}\le -0.\text{5}$ and $0.\text{5}\le \text{x}\le \text{1}.\text{25}$