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Introduction

When analyzing the behavior of a function, it's essential to understand the characteristics of its graph. In this case, we're given the cubic function $f(x)=2x^3-26x-24$. Our goal is to determine which table accurately describes the behavior of the graph of this function.

Analyzing the Function

Before we can determine the behavior of the graph, we need to analyze the function itself. The given function is a cubic function, which means it has a degree of 3. This implies that the graph of the function will have a cubic shape, with three turning points.

Finding the Critical Points

To understand the behavior of the graph, we need to find the critical points of the function. Critical points occur where the derivative of the function is equal to zero or undefined. Let's find the derivative of the function:

f′(x)=6x2−26f'(x) = 6x^2 - 26

Now, we need to set the derivative equal to zero and solve for x:

6x2−26=06x^2 - 26 = 0

6x2=266x^2 = 26

x2=266x^2 = \frac{26}{6}

x2=133x^2 = \frac{13}{3}

x=±133x = \pm \sqrt{\frac{13}{3}}

So, the critical points of the function are $x = \pm \sqrt{\frac{13}{3}}$.

Determining the Behavior of the Graph

Now that we have the critical points, we can determine the behavior of the graph. To do this, we need to examine the intervals between the critical points.

Interval 1: (−∞,−133)(-\infty, -\sqrt{\frac{13}{3}})

In this interval, the derivative of the function is negative, which means the function is decreasing. Since the function is decreasing and the critical point is to the right, the graph will be below the x-axis.

Interval 2: (−133,−3)(-\sqrt{\frac{13}{3}}, -3)

In this interval, the derivative of the function is positive, which means the function is increasing. Since the function is increasing and the critical point is to the left, the graph will be above the x-axis.

Interval 3: (−3,133)(-3, \sqrt{\frac{13}{3}})

In this interval, the derivative of the function is negative, which means the function is decreasing. Since the function is decreasing and the critical point is to the right, the graph will be below the x-axis.

Interval 4: (133,∞)(\sqrt{\frac{13}{3}}, \infty)

In this interval, the derivative of the function is positive, which means the function is increasing. Since the function is increasing and the critical point is to the left, the graph will be above the x-axis.

Conclusion

Based on our analysis, we can conclude that the graph of the function $f(x)=2x^3-26x-24$ will be above the x-axis in the intervals (−133,−3)(-\sqrt{\frac{13}{3}}, -3) and (133,∞)(\sqrt{\frac{13}{3}}, \infty), and below the x-axis in the intervals (−∞,−133)(-\infty, -\sqrt{\frac{13}{3}}) and (−3,133)(-3, \sqrt{\frac{13}{3}}).

Which Table Describes the Behavior of the Graph?

Based on our analysis, we can conclude that the table that accurately describes the behavior of the graph of the function $f(x)=2x^3-26x-24$ is:

Interval Relation of graph to x-axis
(−∞,−3)(-\infty, -3) Above
(−3,133)(-3, \sqrt{\frac{13}{3}}) Below
(133,∞)(\sqrt{\frac{13}{3}}, \infty) Above

This table accurately describes the behavior of the graph of the function $f(x)=2x^3-26x-24$.

Discussion

The behavior of the graph of a function is a crucial aspect of understanding the function itself. By analyzing the function and determining the critical points, we can understand the behavior of the graph. In this case, we found that the graph of the function $f(x)=2x^3-26x-24$ will be above the x-axis in the intervals (−133,−3)(-\sqrt{\frac{13}{3}}, -3) and (133,∞)(\sqrt{\frac{13}{3}}, \infty), and below the x-axis in the intervals (−∞,−133)(-\infty, -\sqrt{\frac{13}{3}}) and (−3,133)(-3, \sqrt{\frac{13}{3}}).

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Calculus, 2nd edition, James Stewart
    Q&A: Understanding the Behavior of the Graph of a Cubic Function ====================================================================

Introduction

In our previous article, we analyzed the behavior of the graph of the cubic function $f(x)=2x^3-26x-24$. We determined the critical points of the function and examined the intervals between them to understand the behavior of the graph. In this article, we'll answer some frequently asked questions about the behavior of the graph of a cubic function.

Q: What is a cubic function?

A: A cubic function is a polynomial function of degree 3, which means it has a cubic shape. The general form of a cubic function is $f(x) = ax^3 + bx^2 + cx + d$, where a, b, c, and d are constants.

Q: What are the critical points of a cubic function?

A: The critical points of a cubic function are the values of x where the derivative of the function is equal to zero or undefined. These points are important because they determine the behavior of the graph of the function.

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

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

Q: What is the behavior of the graph of a cubic function?

A: The behavior of the graph of a cubic function depends on the intervals between the critical points. In each interval, the graph of the function will be either above or below the x-axis.

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

A: To determine the behavior of the graph of a cubic function, you need to examine the intervals between the critical points. In each interval, you need to determine whether the derivative of the function is positive or negative. If the derivative is positive, the graph will be above the x-axis. If the derivative is negative, the graph will be below the x-axis.

Q: What are the intervals between the critical points?

A: The intervals between the critical points are the regions between the critical points where the graph of the function is either above or below the x-axis.

Q: How do I determine the intervals between the critical points?

A: To determine the intervals between the critical points, you need to examine the critical points and determine the intervals between them. In each interval, you need to determine whether the derivative of the function is positive or negative.

Q: What is the significance of the intervals between the critical points?

A: The intervals between the critical points are important because they determine the behavior of the graph of the function. In each interval, the graph of the function will be either above or below the x-axis.

Q: How do I use the intervals between the critical points to understand the behavior of the graph of a cubic function?

A: To use the intervals between the critical points to understand the behavior of the graph of a cubic function, you need to examine the intervals and determine the behavior of the graph in each interval. In each interval, you need to determine whether the derivative of the function is positive or negative. If the derivative is positive, the graph will be above the x-axis. If the derivative is negative, the graph will be below the x-axis.

Conclusion

In this article, we answered some frequently asked questions about the behavior of the graph of a cubic function. We discussed the critical points of a cubic function, the intervals between the critical points, and how to determine the behavior of the graph of a cubic function. We hope this article has been helpful in understanding the behavior of the graph of a cubic function.

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Calculus, 2nd edition, James Stewart