Understanding Increasing and Decreasing Functions

Oct 13, 2025 by windailys.com 11 views

Introduction

In calculus, the study of functions and their behavior is crucial in understanding various mathematical concepts. One of the fundamental concepts in calculus is the study of the increasing and decreasing nature of functions. In this article, we will focus on determining the interval(s) where the function $f(x)=2x^3-3x^2-12x+20$ is only increasing.

Understanding Increasing and Decreasing Functions

To determine the increasing and decreasing nature of a function, we need to find the critical points of the function. Critical points are the points where the function changes from increasing to decreasing or vice versa. The critical points of a function can be found by taking the derivative of the function and setting it equal to zero.

Finding the Derivative of the Function

The derivative of the function $f(x)=2x^3-3x^2-12x+20$ is given by:

f'(x)=6x^2-6x-12

Finding the Critical Points

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

6x^2-6x-12=0

We can solve this quadratic equation using the quadratic formula:

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

where a=6, b=-6, and c=-12.

Plugging in the values, we get:

x=\frac{-(-6)\pm\sqrt{(-6)^2-4(6)(-12)}}{2(6)}

x=\frac{6\pm\sqrt{36+288}}{12}

x=\frac{6\pm\sqrt{324}}{12}

x=\frac{6\pm18}{12}

Solving for x, we get two critical points:

x_1=\frac{6+18}{12}=\frac{24}{12}=2

x_2=\frac{6-18}{12}=\frac{-12}{12}=-1

Determining the Increasing and Decreasing Nature of the Function

To determine the increasing and decreasing nature of the function, we need to examine the sign of the derivative in the intervals defined by the critical points.

For $x<-1$ , the derivative is negative, indicating that the function is decreasing in this interval.

For $-1<x<2$ , the derivative is positive, indicating that the function is increasing in this interval.

For $x>2$ , the derivative is negative, indicating that the function is decreasing in this interval.

Conclusion

Based on the analysis, we can conclude that the function $f(x)=2x^3-3x^2-12x+20$ is only increasing in the interval $-1<x<2$ .

Answer

The correct answer is:

A. $-1\ \textless \ x\ \textless \ 2$

Discussion

The study of increasing and decreasing functions is crucial in understanding various mathematical concepts. In this article, we focused on determining the interval(s) where the function $f(x)=2x^3-3x^2-12x+20$ is only increasing. We found that the function is only increasing in the interval $-1<x<2$ . This analysis can be applied to various real-world problems, such as optimization and modeling.

References

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

Note

Introduction

In our previous article, we discussed how to determine the interval(s) where the function $f(x)=2x^3-3x^2-12x+20$ is only increasing. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights.

Q: What is the significance of finding the critical points of a function?

A: Finding the critical points of a function is crucial in determining the increasing and decreasing nature of the function. Critical points are the points where the function changes from increasing to decreasing or vice versa.

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

A: To find the critical points, we need to take the derivative of the function and set it equal to zero. We can then solve for x to find the critical points.

Q: What is the derivative of the function $f(x)=2x^3-3x^2-12x+20$ ?

A: The derivative of the function $f(x)=2x^3-3x^2-12x+20$ is given by:

f'(x)=6x^2-6x-12

Q: How do you determine the increasing and decreasing nature of a function?

A: To determine the increasing and decreasing nature of a function, we need to examine the sign of the derivative in the intervals defined by the critical points.

Q: What is the significance of the intervals defined by the critical points?

A: The intervals defined by the critical points are crucial in determining the increasing and decreasing nature of the function. By examining the sign of the derivative in these intervals, we can determine where the function is increasing or decreasing.

Q: What is the correct answer to the problem?

A: Based on the analysis, we can conclude that the function $f(x)=2x^3-3x^2-12x+20$ is only increasing in the interval $-1<x<2$ .

Q: Can you provide an example of how to apply this analysis to a real-world problem?

A: Yes, consider a company that wants to maximize its profits by producing a certain product. The company's profit function can be represented by a function $f(x)$ , where x is the number of units produced. By analyzing the increasing and decreasing nature of the function, the company can determine the optimal number of units to produce in order to maximize its profits.

Q: What are some common mistakes to avoid when analyzing the increasing and decreasing nature of a function?

A: Some common mistakes to avoid when analyzing the increasing and decreasing nature of a function include:

Not considering the critical points of the function
Not examining the sign of the derivative in the intervals defined by the critical points
Not considering the domain of the function

Conclusion

In this Q&A article, we provided additional insights and clarification on the analysis of the increasing and decreasing nature of the function $f(x)=2x^3-3x^2-12x+20$ . We hope that this article has been helpful in providing a better understanding of the topic.

References

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

Note

The analysis in this article is based on the assumption that the function is continuous and differentiable in the given interval.