Which Table Describes The Behavior Of The Graph Of $f(x)=2x^3-26x-24$?$[ \begin{array}{|c|c|} \hline \text{Interval} & \text{Relation Of Graph To } X\text{-axis} \ \hline (-\infty, -3) & \text{Above} \ \hline (-3, -1) & \text{Below}

Introduction

In mathematics, the behavior of a function's graph is a crucial aspect of understanding its properties and characteristics. The graph of a function can be described in terms of its relation to the x-axis, which can be above, below, or intersecting the x-axis. In this article, we will analyze the behavior of the graph of the cubic function $f(x)=2x^3-26x-24$ and determine which table describes its behavior.

The Cubic Function

The given cubic function is $f(x)=2x^3-26x-24$. To understand its behavior, we need to find the critical points, which are the points where the function changes from increasing to decreasing or vice versa. We can find the critical points by taking the derivative of the function and setting it equal to zero.

Finding the Derivative

The derivative of the function $f(x)=2x^3-26x-24$ is given by:

$$f'(x)=6x^2-26$$

Finding the Critical Points

To find the critical points, we set the derivative equal to zero and solve for x:

$$6x^2-26=0$$

$$6x^2=26$$

$$x^2=\frac{26}{6}$$

$$x^2=\frac{13}{3}$$

$$x=\pm\sqrt{\frac{13}{3}}$$

$$x=\pm\frac{\sqrt{39}}{3}$$

Analyzing the Behavior of the Graph

Now that we have found the critical points, we can analyze the behavior of the graph. We will examine the intervals where the function is increasing or decreasing.

Interval 1: $(-\infty, -\frac{\sqrt{39}}{3})$

In this interval, the function is decreasing. To determine the relation of the graph to the x-axis, we can evaluate the function at a point in this interval.

Let's evaluate the function at $x=-4$:

$$f(-4)=2(-4)^3-26(-4)-24$$

$$f(-4)=-128+104-24$$

$$f(-4)=-48$$

Since the function is negative, the graph is below the x-axis in this interval.

Interval 2: $(-\frac{\sqrt{39}}{3}, -\frac{\sqrt{39}}{3})$

In this interval, the function is increasing. To determine the relation of the graph to the x-axis, we can evaluate the function at a point in this interval.

Let's evaluate the function at $x=-\frac{\sqrt{39}}{3}$:

$$f(-\frac{\sqrt{39}}{3})=2(-\frac{\sqrt{39}}{3})^3-26(-\frac{\sqrt{39}}{3})-24$$

$$f(-\frac{\sqrt{39}}{3})=-\frac{26\sqrt{39}}{9}+\frac{26\sqrt{39}}{3}-24$$

$$f(-\frac{\sqrt{39}}{3})=-\frac{26\sqrt{39}}{9}+\frac{78\sqrt{39}}{9}-24$$

$$f(-\frac{\sqrt{39}}{3})=\frac{52\sqrt{39}}{9}-24$$

Since the function is positive, the graph is above the x-axis in this interval.

Interval 3: $(-\frac{\sqrt{39}}{3}, \frac{\sqrt{39}}{3})$

In this interval, the function is decreasing. To determine the relation of the graph to the x-axis, we can evaluate the function at a point in this interval.

Let's evaluate the function at $x=0$:

$$f(0)=2(0)^3-26(0)-24$$

$$f(0)=-24$$

Since the function is negative, the graph is below the x-axis in this interval.

Interval 4: $(\frac{\sqrt{39}}{3}, \infty)$

In this interval, the function is increasing. To determine the relation of the graph to the x-axis, we can evaluate the function at a point in this interval.

Let's evaluate the function at $x=4$:

$$f(4)=2(4)^3-26(4)-24$$

$$f(4)=128-104-24$$

$$f(4)=0$$

Since the function is zero, the graph intersects the x-axis in this interval.

Conclusion

In conclusion, the graph of the cubic function $f(x)=2x^3-26x-24$ is above the x-axis in the interval $(-\infty, -3)$, below the x-axis in the interval $(-3, -1)$, and intersects the x-axis in the interval $(1, \infty)$. The table that describes the behavior of the graph is:

Interval	Relation of graph to x-axis
$(-\infty, -3)$	Above
$(-3, -1)$	Below
$(-1, \infty)$	Above

$$ $$

Q: What is the purpose of analyzing the behavior of a function's graph?

A: Analyzing the behavior of a function's graph is crucial in understanding its properties and characteristics. It helps us determine the intervals where the function is increasing or decreasing, which is essential in solving optimization problems, finding the maximum or minimum values of the function, and understanding the behavior of the function as it approaches certain points.

Q: How do I determine the relation of the graph to the x-axis?

A: To determine the relation of the graph to the x-axis, you need to evaluate the function at a point in the interval. If the function is positive, the graph is above the x-axis. If the function is negative, the graph is below the x-axis. If the function is zero, the graph intersects the x-axis.

Q: What is the significance of the critical points in analyzing the behavior of the graph?

A: Critical points are the points where the function changes from increasing to decreasing or vice versa. They are significant in analyzing the behavior of the graph because they help us determine the intervals where the function is increasing or decreasing.

Q: How do I find the critical points of a function?

A: To find the critical points of a function, you need to take the derivative of the function and set it equal to zero. Then, you need to solve for x to find the critical points.

Q: What is the difference between a local maximum and a global maximum?

A: A local maximum is a point where the function has a maximum value within a certain interval. A global maximum is a point where the function has a maximum value over its entire domain.

Q: How do I determine if a point is a local maximum or a global maximum?

A: To determine if a point is a local maximum or a global maximum, you need to evaluate the function at the point and compare it to the function's values at other points in the interval. If the function has a higher value at the point than at any other point in the interval, it is a local maximum. If the function has a higher value at the point than at any other point in its entire domain, it is a global maximum.

Q: What is the significance of the intervals in analyzing the behavior of the graph?

A: Intervals are significant in analyzing the behavior of the graph because they help us determine the relation of the graph to the x-axis and the behavior of the function within each interval.

Q: How do I determine the intervals where the function is increasing or decreasing?

A: To determine the intervals where the function is increasing or decreasing, you need to evaluate the derivative of the function at a point in the interval. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.

Q: What is the difference between a cubic function and a quadratic function?

A: A cubic function is a function of the form $f(x)=ax^3+bx^2+cx+d$, where a, b, c, and d are constants. A quadratic function is a function of the form $f(x)=ax^2+bx+c$, where a, b, and c are constants.

Q: How do I determine the behavior of a cubic function?

A: To determine the behavior of a cubic function, you need to find the critical points of the function and evaluate the function at a point in each interval. Then, you need to compare the function's values at each point to determine the relation of the graph to the x-axis.

Conclusion

In conclusion, analyzing the behavior of a function's graph is a crucial aspect of understanding its properties and characteristics. By determining the relation of the graph to the x-axis and the behavior of the function within each interval, you can gain a deeper understanding of the function's behavior and make informed decisions about its application.