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
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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 4.3.17
Identifying Extrema
In Exercises 15–18:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local and absolute extreme values, if any, saying where they occur.
Verified step by step guidance1
Observe the graph of the function y = f(x) and identify the critical points where the function changes direction. These points are typically where the derivative is zero or undefined.
Determine the intervals where the function is increasing. A function is increasing on an interval if its derivative is positive over that interval. From the graph, identify the segments where the slope of the function is positive.
Determine the intervals where the function is decreasing. A function is decreasing on an interval if its derivative is negative over that interval. From the graph, identify the segments where the slope of the function is negative.
Identify the local extrema by examining the critical points. A local maximum occurs where the function changes from increasing to decreasing, and a local minimum occurs where the function changes from decreasing to increasing.
Identify the absolute extrema by comparing the values of the function at the critical points and endpoints of the interval. The highest value is the absolute maximum, and the lowest value is the absolute minimum.
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Video duration:
6mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Increasing and Decreasing Intervals
A function is increasing on an interval if, for any two points within the interval, a larger x-value results in a larger y-value. Conversely, it is decreasing if a larger x-value results in a smaller y-value. To determine these intervals, one can analyze the graph's slope or use the first derivative test, where a positive derivative indicates increasing and a negative derivative indicates decreasing.
Recommended video:
07:32
Determining Where a Function is Increasing & Decreasing
Local and Absolute Extrema
Local extrema are points where a function reaches a local maximum or minimum within a neighborhood, while absolute extrema are the highest or lowest points over the entire domain. Local extrema occur where the derivative changes sign, and absolute extrema can be found by evaluating the function at critical points and endpoints. The graph helps visualize these points as peaks (maxima) or troughs (minima).
Recommended video:
05:58Finding Extrema Graphically
Critical Points
Critical points of a function occur where its derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection. These points are crucial for identifying intervals of increase or decrease and locating extrema. By analyzing the graph or calculating the derivative, one can pinpoint these critical points and assess their nature using the first or second derivative tests.
Recommended video:
04:50Critical Points
Related Practice
Textbook Question
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
d²r/dt² = 2/t³; dr/dt|ₜ ₌ ₁ =1, r(1) = 1
Textbook Question
113. If b, c, and d are constants, for what value of b will the curve y = x^3 + bx^2 + cx + d have a
point of inflection at x = 1? Give reasons for your answer.
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
Checking the Mean Value Theorem
Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.
g(x) = {x³, −2 ≤ x ≤ 0
x², 0 < x ≤ 2
Textbook Question
37. What value of a makes f(x) = x^2 +(a/x) have
a. a local minimum at x = 2?
b. a point of inflection at x = 1?