Understanding the Basics of Increasing and Decreasing Functions

Oct 13, 2025 by windailys.com 117 views

Let's Dive into the World of Increasing and Decreasing Functions

Understanding the Basics of Increasing and Decreasing Functions

Increasing and decreasing functions are fundamental concepts in calculus that help us understand the behavior of functions. In this article, we will explore the concept of increasing and decreasing functions, and we will use the given function $$f(x) = 4x^3 - 48x + 4$$ to illustrate these concepts.

What are Increasing and Decreasing Functions?

An increasing function is a function that always increases as the input value increases. In other words, if we have a function $$f(x)$$ and we know that $$f(x_1) < f(x_2)$$ whenever $$x_1 < x_2$$, then we say that $$f(x)$$ is an increasing function.

On the other hand, a decreasing function is a function that always decreases as the input value increases. In other words, if we have a function $$f(x)$$ and we know that $$f(x_1) > f(x_2)$$ whenever $$x_1 < x_2$$, then we say that $$f(x)$$ is a decreasing function.

Finding the Intervals on Which $$f$$ is Increasing

To find the intervals on which $$f$$ is increasing, we need to find the critical points of the function. The critical points are the points where the derivative of the function is equal to zero or undefined.

Let's find the derivative of the function $$f(x) = 4x^3 - 48x + 4$$.

f'(x) = 12x^2 - 48

Now, let's set the derivative equal to zero and solve for $$x$$.

12x^2 - 48 = 0

x^2 = 4

x = \pm 2

So, the critical points are $$x = -2$$ and $$x = 2$$.

Now, let's use the first derivative test to determine the intervals on which $$f$$ is increasing.

If the derivative is positive, then the function is increasing. If the derivative is negative, then the function is decreasing.

Let's evaluate the derivative at a point to the left of $$x = -2$$.

f'(-3) = 12(-3)^2 - 48

f'(-3) = 108 - 48

f'(-3) = 60

Since the derivative is positive, the function is increasing on the interval $$(-\infty, -2)$$.

Let's evaluate the derivative at a point to the right of $$x = -2$$.

f'(-1) = 12(-1)^2 - 48

f'(-1) = 12 - 48

f'(-1) = -36

Since the derivative is negative, the function is decreasing on the interval $$(-2, 2)$$.

Let's evaluate the derivative at a point to the right of $$x = 2$$.

f'(3) = 12(3)^2 - 48

f'(3) = 108 - 48

f'(3) = 60

Since the derivative is positive, the function is increasing on the interval $$(2, \infty)$$.

Finding the Intervals on Which $$f$$ is Decreasing

To find the intervals on which $$f$$ is decreasing, we need to find the critical points of the function. The critical points are the points where the derivative of the function is equal to zero or undefined.

We already found the critical points in the previous section. The critical points are $$x = -2$$ and $$x = 2$$.

Now, let's use the first derivative test to determine the intervals on which $$f$$ is decreasing.

If the derivative is negative, then the function is decreasing. If the derivative is positive, then the function is increasing.

Let's evaluate the derivative at a point to the left of $$x = -2$$.

f'(-3) = 12(-3)^2 - 48

f'(-3) = 108 - 48

f'(-3) = 60

Since the derivative is positive, the function is increasing on the interval $$(-\infty, -2)$$.

Let's evaluate the derivative at a point to the right of $$x = -2$$.

f'(-1) = 12(-1)^2 - 48

f'(-1) = 12 - 48

f'(-1) = -36

Since the derivative is negative, the function is decreasing on the interval $$(-2, 2)$$.

Let's evaluate the derivative at a point to the right of $$x = 2$$.

f'(3) = 12(3)^2 - 48

f'(3) = 108 - 48

f'(3) = 60

Since the derivative is positive, the function is increasing on the interval $$(2, \infty)$$.

Conclusion

In this article, we explored the concept of increasing and decreasing functions. We used the given function $$f(x) = 4x^3 - 48x + 4$$ to illustrate these concepts. We found the intervals on which $$f$$ is increasing and decreasing, and we used the first derivative test to determine these intervals.

Final Answer

The final answer is:

The function $$f$$ is increasing on the intervals $$(-\infty, -2)$$ and $$(2, \infty)$$.
The function $$f$$ is decreasing on the interval $$(-2, 2)$$.

Note

The final answer is in the format of a list of intervals. The first item in the list corresponds to the interval on which the function is increasing, and the second item corresponds to the interval on which the function is decreasing.
Q&A: Increasing and Decreasing Functions

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about increasing and decreasing functions.

Q: What is the difference between an increasing function and a decreasing function?

A: An increasing function is a function that always increases as the input value increases. On the other hand, a decreasing function is a function that always decreases as the input value increases.

Q: How do I determine if a function is increasing or decreasing?

A: To determine if a function is increasing or decreasing, you need to find the derivative of the function. If the derivative is positive, then the function is increasing. If the derivative is negative, then the function is decreasing.

Q: What are the critical points of a function?

A: The critical points of a function are the points where the derivative of the function is equal to zero or undefined. These points are important because they can help you determine the intervals on which the function is increasing or decreasing.

Q: How do I use the first derivative test to determine if a function is increasing or decreasing?

A: To use the first derivative test, you need to evaluate the derivative of the function at a point to the left of the critical point and a point to the right of the critical point. If the derivative is positive at both points, then the function is increasing on the interval between the two points. If the derivative is negative at both points, then the function is decreasing on the interval between the two points.

Q: What is the significance of the first derivative test?

A: The first derivative test is a powerful tool that helps you determine the intervals on which a function is increasing or decreasing. It is based on the idea that if the derivative of a function is positive, then the function is increasing, and if the derivative is negative, then the function is decreasing.

Q: Can a function be both increasing and decreasing at the same time?

A: No, a function cannot be both increasing and decreasing at the same time. If a function is increasing on an interval, then it is decreasing on the interval between the critical points. If a function is decreasing on an interval, then it is increasing on the interval between the critical points.

Q: How do I find the intervals on which a function is increasing or decreasing?

A: To find the intervals on which a function is increasing or decreasing, you need to find the critical points of the function and use the first derivative test to determine the intervals on which the function is increasing or decreasing.

Q: What is the relationship between the derivative of a function and the function itself?

A: The derivative of a function is a measure of how fast the function changes as the input value changes. If the derivative is positive, then the function is increasing. If the derivative is negative, then the function is decreasing.

Q: Can a function have multiple intervals on which it is increasing or decreasing?

A: Yes, a function can have multiple intervals on which it is increasing or decreasing. For example, a function can be increasing on one interval and decreasing on another interval.

Q: How do I determine if a function is increasing or decreasing on a specific interval?

A: To determine if a function is increasing or decreasing on a specific interval, you need to find the critical points of the function and use the first derivative test to determine the intervals on which the function is increasing or decreasing.

Conclusion

In this article, we answered some of the most frequently asked questions about increasing and decreasing functions. We hope that this article has helped you understand the concept of increasing and decreasing functions and how to use the first derivative test to determine the intervals on which a function is increasing or decreasing.

Final Answer

The final answer is:

An increasing function is a function that always increases as the input value increases.
A decreasing function is a function that always decreases as the input value increases.
The first derivative test is a powerful tool that helps you determine the intervals on which a function is increasing or decreasing.
The critical points of a function are the points where the derivative of the function is equal to zero or undefined.
The first derivative test is based on the idea that if the derivative of a function is positive, then the function is increasing, and if the derivative is negative, then the function is decreasing.
A function cannot be both increasing and decreasing at the same time.
The derivative of a function is a measure of how fast the function changes as the input value changes.
A function can have multiple intervals on which it is increasing or decreasing.
To determine if a function is increasing or decreasing on a specific interval, you need to find the critical points of the function and use the first derivative test to determine the intervals on which the function is increasing or decreasing.