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) = x(x − 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 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 4.1.52c
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
b. Does f'(-3) exist?
Verified step by step guidance1
To determine if f'(-3) exists, we need to check if the function f(x) = |x³ − 9x| is differentiable at x = -3. Differentiability requires the function to be continuous and have a defined derivative at that point.
First, check the continuity of f(x) at x = -3. Since f(x) is an absolute value function, it is continuous everywhere, including at x = -3.
Next, consider the definition of the derivative. The derivative f'(x) exists at x = -3 if the limit of the difference quotient exists as x approaches -3 from both sides.
To find the derivative, consider the piecewise nature of the absolute value function. The expression inside the absolute value, x³ - 9x, changes sign at the roots of the equation x³ - 9x = 0. Solve for x to find these critical points.
Evaluate the derivative from the left and right of x = -3 using the piecewise definition of f(x). If the left-hand and right-hand derivatives are equal, then f'(-3) exists. Otherwise, it does not.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is the slope of the tangent line to the function's graph at that point. For a function f(x), the derivative is denoted as f'(x) and is found using differentiation rules.
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Derivatives
Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x. It affects the differentiability of functions because it creates sharp corners or cusps in graphs, where the derivative may not exist. For f(x) = |x³ − 9x|, the absolute value impacts the function's smoothness and continuity.
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06:37Average Value of a Function
Differentiability and Continuity
A function is differentiable at a point if it is smooth and has a defined tangent at that point, implying continuity. However, a function can be continuous but not differentiable at points where there are sharp turns or cusps, such as those introduced by absolute values. Checking differentiability involves examining the function's behavior around the point of interest.
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05:34Intro to Continuity
Related Practice
Textbook Question
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫tanθ sec²θ dθ = (1/2) sec²θ + C
2
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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.
-sec²(3x/2)
1
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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) = (x − 1)²(x + 2)
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.
−π csc (πx/2) cot (πx/2)
1
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Textbook Question
Finding displacement from an antiderivative of velocity
a. Suppose that the velocity of a body moving along the s-axis is
ds/dt = v = 9.8t − 3.
iii. Now find the body’s displacement from t = 1 to t = 3 given that s = s₀ when t = 0.