Textbook Question
Applications
Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).
Find:
∫[f(x) + g(x)] dx
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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 4.1.52d
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
d. Determine all extrema of f.
Verified step by step guidance1
To find the extrema of the function \( f(x) = |x^3 - 9x| \), we first need to consider the critical points of the function. Critical points occur where the derivative is zero or undefined.
First, find the derivative of the inside function \( g(x) = x^3 - 9x \). The derivative is \( g'(x) = 3x^2 - 9 \).
Set \( g'(x) = 0 \) to find the critical points of \( g(x) \). Solve \( 3x^2 - 9 = 0 \) to find the values of \( x \) where the derivative is zero.
The absolute value function \( f(x) = |g(x)| \) can change its behavior at points where \( g(x) = 0 \). Solve \( x^3 - 9x = 0 \) to find these points.
Evaluate \( f(x) \) at the critical points and endpoints of the domain to determine the extrema. Consider the behavior of \( f(x) \) around these points to classify them as local minima, maxima, or neither.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value Function
The absolute value function, denoted as |x|, represents the distance of a number x from zero on the number line, always yielding a non-negative result. In the context of f(x) = |x³ − 9x|, it affects how we find extrema by considering both the positive and negative scenarios of the expression inside the absolute value.
Recommended video:
06:37
Average Value of a Function
Critical Points
Critical points of a function occur where its derivative is zero or undefined. These points are potential candidates for local extrema. For f(x) = |x³ − 9x|, we first need to find the derivative, set it to zero, and solve for x to identify critical points, considering the behavior of the absolute value function.
Recommended video:
04:50Critical Points
Extrema
Extrema refer to the maximum and minimum values of a function. To determine extrema, evaluate the function at critical points and endpoints of the domain. For f(x) = |x³ − 9x|, analyze the critical points and the behavior of the function as x approaches positive and negative infinity to identify all local and global extrema.
Recommended video:
05:58Finding Extrema Graphically
Related Practice
Textbook Question
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
y=1-(x+1)^3
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Textbook Question
105. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (c) Acceleration equal to zero?
193
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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:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = 1− 4/x², x ≠ 0
120
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Textbook Question
Applications
Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).
Find:
∫[−f(x)] dx
19
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Textbook Question
106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (d) When is the acceleration positive? Negative?
217
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