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
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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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.1.51b
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = (x − 2)²ᐟ³.
b. Show that the only local extreme value of f occurs at x = 2.
Verified step by step guidance1
To find local extreme values, we need to determine where the derivative of the function f(x) = (x - 2)^(2/3) is zero or undefined. Start by finding the derivative f'(x).
Use the chain rule to differentiate f(x). Let u = x - 2, then f(x) = u^(2/3). The derivative f'(x) = (2/3) * u^(-1/3) * du/dx, where du/dx = 1.
Simplify the derivative: f'(x) = (2/3) * (x - 2)^(-1/3). This derivative is undefined at x = 2 because the expression (x - 2)^(-1/3) involves division by zero.
Since f'(x) is undefined at x = 2, check the behavior of f(x) around x = 2 to determine if it is a local extreme value. Consider the sign of f'(x) as x approaches 2 from the left and right.
Observe that as x approaches 2 from the left, f'(x) is negative, and as x approaches 2 from the right, f'(x) is positive. This change in sign indicates that x = 2 is a local minimum.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Local Extreme Values
Local extreme values refer to points in the domain of a function where the function takes on a maximum or minimum value relative to nearby points. To identify these points, we typically use the first derivative test, which involves finding critical points where the derivative is zero or undefined, and then analyzing the behavior of the function around these points.
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Average Value of a Function
First Derivative Test
The first derivative test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By evaluating the sign of the derivative before and after the critical point, we can conclude if the function is increasing or decreasing, thus identifying the nature of the extreme value at that point.
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07:09The First Derivative Test: Finding Local Extrema
Critical Points
Critical points of a function occur where its derivative is either zero or undefined. These points are essential for finding local extreme values, as they represent potential locations where the function's behavior changes. In the context of the given function f(x) = (x − 2)²/3, identifying critical points will help in determining where the local extreme value occurs.
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04:50Critical Points
Related Practice
Textbook Question
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫3(2x + 1)² dx = (2x + 1)³ + C
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Textbook Question
Identifying Extrema
In Exercises 19–40:
b. Identify the function’s local extreme values, if any, saying where they occur.
f(r) = 3r³ + 16r
132
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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.
4 sec 3x tan 3x
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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
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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)
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