Skip to main content
Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.3.9b
Chapter 4, Problem 4.3.9b

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:


b. On what open intervals is f increasing or decreasing?


f′(x) = 1− 4/x², x ≠ 0

Verified step by step guidance
1
Identify the critical points by setting the derivative \( f'(x) = 1 - \frac{4}{x^2} \) equal to zero and solving for \( x \). This will help determine where the function changes from increasing to decreasing or vice versa.
Solve the equation \( 1 - \frac{4}{x^2} = 0 \) to find the critical points. Rearrange the equation to \( \frac{4}{x^2} = 1 \) and solve for \( x \).
Determine the sign of \( f'(x) \) on the intervals defined by the critical points. Choose test points in each interval and substitute them into \( f'(x) \) to see if the derivative is positive or negative.
If \( f'(x) > 0 \) on an interval, then \( f(x) \) is increasing on that interval. If \( f'(x) < 0 \), then \( f(x) \) is decreasing on that interval.
Summarize the intervals where \( f(x) \) is increasing and decreasing based on the sign of \( f'(x) \) in each interval.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative and Function Behavior

The derivative of a function, f'(x), provides information about the function's rate of change. If f'(x) > 0 on an interval, the function is increasing there; if f'(x) < 0, the function is decreasing. Understanding the sign of the derivative is crucial for determining where the function is increasing or decreasing.
Recommended video:
06:30
Derivatives of Other Trig Functions

Critical Points and Intervals

Critical points occur where the derivative is zero or undefined, indicating potential changes in the function's behavior. To find intervals of increase or decrease, identify these points and test the sign of the derivative in the intervals they create. This helps in understanding the function's behavior across its domain.
Recommended video:
04:50
Critical Points

Open Intervals

Open intervals are ranges of x-values where the function's behavior is consistent, either increasing or decreasing. They do not include their endpoints, which is important when analyzing functions, especially when the derivative is undefined at certain points, like x = 0 in this case.
Recommended video:
02:59
Finding Area Between Curves that Cross on the Interval
Related Practice
Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫xsinx dx = -x cos x + C

1
views
Textbook Question

Identifying Extrema


In Exercises 19–40:


b. Identify the function’s local extreme values, if any, saying where they occur.


f(θ) = 3θ² − 4θ³

Textbook Question

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = |x³ − 9x|.


b. Does f'(3) exist?

Textbook Question

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:



b. On what open intervals is f increasing or decreasing?


f′(x) = (x − 1)(x + 2)(x − 3)

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

x⁻³/2 + x²

1
views
Textbook Question

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:



b. On what open intervals is f increasing or decreasing?


f′(x) = (x − 1)²(x + 2)