Determining Intervals of Increase and Decrease

Oct 13, 2025 by leolavanderia.com.br 68 views

Introduction

In calculus, the concept of increasing and decreasing functions is crucial in understanding the behavior of a function. Given a function $$f(x)$$, we want to determine the intervals on which the function is increasing or decreasing. This involves finding the critical points of the function, which are the points where the function changes from increasing to decreasing or vice versa.

Finding Critical Points

To find the critical points of the function $$f(x) = 4x^3 - 48x + 4$$, we need to find the derivative of the function and set it equal to zero. The derivative of $$f(x)$$ is given by:

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

Setting the derivative equal to zero, we get:

12x^2 - 48 = 0

Solving for $$x$$, we get:

x^2 = 4

x = \pm 2

Therefore, the critical points of the function are $$x = -2$$ and $$x = 2$$.

Determining Intervals of Increase and Decrease

To determine the intervals on which the function is increasing or decreasing, we need to examine the sign of the derivative $$f'(x)$$ on either side of the critical points.

Interval $(-\infty, -2)$

On the interval $(-\infty, -2)$ , we can choose a test point, say $$x = -3$$. Plugging this value into the derivative, we get:

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

f'(-3) = 12(9) - 48

f'(-3) = 108 - 48

f'(-3) = 60

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

Interval $(-2, 2)$

On the interval $(-2, 2)$ , we can choose a test point, say $$x = 0$$. Plugging this value into the derivative, we get:

f'(0) = 12(0)^2 - 48

f'(0) = 12(0) - 48

f'(0) = -48

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

Interval $(2, \infty)$

On the interval $(2, \infty)$ , we can choose a test point, say $$x = 3$$. Plugging this value into the derivative, we get:

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

f'(3) = 12(9) - 48

f'(3) = 108 - 48

f'(3) = 60

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

Conclusion

In conclusion, the function $$f(x) = 4x^3 - 48x + 4$$ is increasing on the intervals $(-\infty, -2)$ $ and $$(2, \infty)$$. The function is decreasing on the interval $(-2, 2)$ .

Final Answer

Input the interval(s) on which $$f$$ is increasing: - $(-\infty, -2)$ $ and $$(2, \infty)$$
Input the interval(s) on which $$f$$ is decreasing: - $(-2, 2)$
Find the point(s) at which $$f$$ changes from increasing to decreasing: - $x = -2$ $ and $$x = 2$$

Introduction

In our previous article, we explored the concept of increasing and decreasing functions, and applied it to the function $$f(x) = 4x^3 - 48x + 4$$. We found that the function is increasing on the intervals $(-\infty, -2)$ $ and $$(2, \infty)$$, and decreasing on the interval $(-2, 2)$ .

Q&A

Q: What is the significance of finding the intervals of increase and decrease of a function?

A: Finding the intervals of increase and decrease of a function is crucial in understanding the behavior of the function. It helps us to identify the points where the function changes from increasing to decreasing or vice versa, which is essential in various applications such as optimization, physics, and engineering.

Q: How do we determine the intervals of increase and decrease of a function?

A: To determine the intervals of increase and decrease of a function, we need to find the critical points of the function, 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.

Q: What is the relationship between the derivative of a function and its intervals of increase and decrease?

A: The derivative of a function represents the rate of change of the function. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing. If the derivative is zero, the function is neither increasing nor decreasing.

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

A: No, a function cannot be increasing and decreasing at the same time. However, a function can be increasing on one interval and decreasing on another interval.

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

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

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

A: The critical points of a function are the points where the function changes from increasing to decreasing or vice versa. They are essential in understanding the behavior of the function and are used in various applications such as optimization, physics, and engineering.

Q: Can a function have multiple critical points?

A: Yes, a function can have multiple critical points. In fact, a function can have an infinite number of critical points.

Q: How do we determine the intervals of increase and decrease of a function with multiple critical points?

A: To determine the intervals of increase and decrease of a function with multiple critical points, we need to examine the sign of the derivative on either side of each critical point.

Conclusion

In conclusion, finding the intervals of increase and decrease of a function is crucial in understanding the behavior of the function. We can find the intervals of increase and decrease by taking the derivative of the function and setting it equal to zero. The critical points of a function are the points where the function changes from increasing to decreasing or vice versa, and are essential in understanding the behavior of the function.

Final Answer

Q: What is the significance of finding the intervals of increase and decrease of a function? A: Finding the intervals of increase and decrease of a function is crucial in understanding the behavior of the function.
Q: How do we determine the intervals of increase and decrease of a function? A: To determine the intervals of increase and decrease of a function, we need to find the critical points of the function, which are the points where the function changes from increasing to decreasing or vice versa.
Q: Can a function be increasing and decreasing at the same time? A: No, a function cannot be increasing and decreasing at the same time.