Textbook Question
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = √(x(1 − x)), [0, 1]
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.3.35
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
f(x) = (x² − 3) / (x − 2), x ≠ 2
Verified step by step guidance1
To determine where the function \( f(x) = \frac{x^2 - 3}{x - 2} \) is increasing or decreasing, first find its derivative \( f'(x) \). Use the quotient rule: \( f'(x) = \frac{(x - 2)(2x) - (x^2 - 3)(1)}{(x - 2)^2} \). Simplify the expression to find \( f'(x) \).
Set \( f'(x) = 0 \) to find critical points. Solve the equation \( (x - 2)(2x) - (x^2 - 3) = 0 \) to find the values of \( x \) where the derivative is zero. These are potential points where the function changes from increasing to decreasing or vice versa.
Determine the sign of \( f'(x) \) on the intervals defined by the critical points and the point where the function is undefined (\( x = 2 \)). Choose test points in each interval to evaluate the sign of \( f'(x) \). If \( f'(x) > 0 \), the function is increasing; if \( f'(x) < 0 \), the function is decreasing.
Identify the local extrema by evaluating \( f(x) \) at the critical points found in step 2. A change from positive to negative in \( f'(x) \) indicates a local maximum, while a change from negative to positive indicates a local minimum.
Summarize the intervals of increase and decrease, and list any local extrema along with their locations. Remember to exclude \( x = 2 \) from the domain of the function, as it is undefined there.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative and Critical Points
The derivative of a function provides information about its rate of change. To find where a function is increasing or decreasing, calculate its derivative and identify critical points where the derivative is zero or undefined. These points help determine intervals of increase or decrease by testing the sign of the derivative in each interval.
Recommended video:
04:50
Critical Points
Intervals of Increase and Decrease
Once critical points are identified, determine the intervals on which the function is increasing or decreasing. This involves testing the sign of the derivative in each interval between critical points. A positive derivative indicates the function is increasing, while a negative derivative indicates it is decreasing.
Recommended video:
07:32Determining Where a Function is Increasing & Decreasing
Local Extrema
Local extrema refer to the local maximum or minimum values of a function. These occur at critical points where the derivative changes sign. A change from positive to negative indicates a local maximum, while a change from negative to positive indicates a local minimum. Evaluating the function at these points confirms the extrema.
Recommended video:
05:58Finding Extrema Graphically
Related Practice
Textbook Question
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(−2cost) dt
Textbook Question
Applications
Stopping a motorcycle The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from 30 mph (44 ft/sec) to 0 in 45 ft. What constant deceleration does it take to do that?
Textbook Question
Finding Extrema from Graphs
In Exercises 11–14, match the table with a graph.
1
views
Textbook Question
Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫ cos³ 𝓍/2 d𝓍
Textbook Question
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(t√t + √t) / t² dt