The Function and Its Derivative

Oct 13, 2025 by windailys.com 28 views

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 provide valuable insights into its behavior, including its increasing or decreasing nature, its maximum or minimum values, and its points of inflection. In this article, we will delve into the behavior of the graph of the cubic function $f(x) = 2x^3 - 26x - 24$ and determine which table accurately describes its behavior.

The Function and Its Derivative

The given function is a cubic function, which means it has a degree of 3. The general form of a cubic function is $f(x) = ax^3 + bx^2 + cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants. In this case, the function is $f(x) = 2x^3 - 26x - 24$ . To understand the behavior of the graph, we need to find the derivative of the function, which represents the rate of change of the function with respect to $x$ .

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

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

Critical Points

Critical points are the points on the graph where the function changes from increasing to decreasing or vice versa. To find the critical points, we need to find the values of $x$ where the derivative is equal to zero or undefined.

Setting the derivative equal to zero, we get:

6x^2 - 26 = 0

Solving for $x$ , we get:

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

x = \pm \sqrt{\frac{26}{6}}

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

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

Therefore, the critical points are $x = \frac{\sqrt{39}}{3}$ and $x = -\frac{\sqrt{39}}{3}$ .

Intervals of Increase and Decrease

To determine the intervals of increase and decrease, we need to examine the sign of the derivative in each interval. If the derivative is positive, the function is increasing, and if it is negative, the function is decreasing.

In the interval $(-\infty, -\frac{\sqrt{39}}{3})$ , the derivative is negative, so the function is decreasing.

In the interval $(-\frac{\sqrt{39}}{3}, \frac{\sqrt{39}}{3})$ , the derivative is positive, so the function is increasing.

In the interval $(\frac{\sqrt{39}}{3}, \infty)$ , the derivative is negative, so the function is decreasing.

Points of Inflection

Points of inflection are the points on the graph where the function changes from concave up to concave down or vice versa. To find the points of inflection, we need to find the values of $x$ where the second derivative is equal to zero or undefined.

The second derivative of the function is given by:

f''(x) = 12x

Setting the second derivative equal to zero, we get:

12x = 0

Solving for $x$ , we get:

x = 0

Therefore, the point of inflection is $x = 0$ .

Conclusion

In conclusion, the graph of the function $f(x) = 2x^3 - 26x - 24$ has the following behavior:

In the interval $(-\infty, -3)$ , the graph is above the $x$ -axis.
In the interval $(-3, -\frac{\sqrt{39}}{3})$ , the graph is decreasing.
In the interval $(-\frac{\sqrt{39}}{3}, \frac{\sqrt{39}}{3})$ , the graph is increasing.
In the interval $(\frac{\sqrt{39}}{3}, \infty)$ , the graph is decreasing.
The point of inflection is $x = 0$ .

Therefore, 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
$(-\infty, -3)$	Above
$(-3, -\frac{\sqrt{39}}{3})$	Below
$(-\frac{\sqrt{39}}{3}, \frac{\sqrt{39}}{3})$	Above
$(\frac{\sqrt{39}}{3}, \infty)$	Below

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Calculus, 2nd edition, James Stewart
Frequently Asked Questions: Understanding the Behavior of the Graph of $f(x) = 2x^3 - 26x - 24$ ====================================================================================

Q: What is the degree of the function $f(x) = 2x^3 - 26x - 24$ ?

A: The degree of the function $f(x) = 2x^3 - 26x - 24$ is 3, which means it is a cubic function.

Q: What is the derivative of the function $f(x) = 2x^3 - 26x - 24$ ?

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

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

Q: What are the critical points of the function $f(x) = 2x^3 - 26x - 24$ ?

A: The critical points of the function $f(x) = 2x^3 - 26x - 24$ are $x = \frac{\sqrt{39}}{3}$ and $x = -\frac{\sqrt{39}}{3}$ .

Q: In which intervals is the function $f(x) = 2x^3 - 26x - 24$ increasing or decreasing?

A: The function $f(x) = 2x^3 - 26x - 24$ is:

Decreasing in the interval $(-\infty, -\frac{\sqrt{39}}{3})$
Increasing in the interval $(-\frac{\sqrt{39}}{3}, \frac{\sqrt{39}}{3})$
Decreasing in the interval $(\frac{\sqrt{39}}{3}, \infty)$

Q: What is the point of inflection of the function $f(x) = 2x^3 - 26x - 24$ ?

A: The point of inflection of the function $f(x) = 2x^3 - 26x - 24$ is $x = 0$ .

Q: How does the graph of the function $f(x) = 2x^3 - 26x - 24$ relate to the $x$ -axis in different intervals?

A: The graph of the function $f(x) = 2x^3 - 26x - 24$ is:

Above the $x$ -axis in the interval $(-\infty, -3)$
Below the $x$ -axis in the interval $(-3, -\frac{\sqrt{39}}{3})$
Above the $x$ -axis in the interval $(-\frac{\sqrt{39}}{3}, \frac{\sqrt{39}}{3})$
Below the $x$ -axis in the interval $(\frac{\sqrt{39}}{3}, \infty)$

Q: What is the significance of the critical points and the point of inflection in the behavior of the graph of the function $f(x) = 2x^3 - 26x - 24$ ?

A: The critical points and the point of inflection are significant in understanding the behavior of the graph of the function $f(x) = 2x^3 - 26x - 24$ . They help to identify the intervals where the function is increasing or decreasing and the points where the function changes from concave up to concave down or vice versa.

Q: How can the behavior of the graph of the function $f(x) = 2x^3 - 26x - 24$ be used in real-world applications?

A: The behavior of the graph of the function $f(x) = 2x^3 - 26x - 24$ can be used in various real-world applications, such as:

Modeling population growth or decline
Analyzing the behavior of physical systems, such as the motion of objects under the influence of gravity or friction
Understanding the behavior of economic systems, such as the growth or decline of industries or economies

Q: What are some common mistakes to avoid when analyzing the behavior of the graph of a function?

A: Some common mistakes to avoid when analyzing the behavior of the graph of a function include:

Failing to identify the critical points and the point of inflection
Misinterpreting the sign of the derivative or the second derivative
Failing to consider the intervals where the function is increasing or decreasing
Failing to consider the points where the function changes from concave up to concave down or vice versa.