Textbook Question
110. Suppose the derivative of the function y = f(x) is
y'=(x-1)^22(x-2)(x-4).
At what points, if any, does the graph of f have a local minimum, local maximum, or
point of inflection?
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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 4.4.7
Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.
7. y=sin|x|, -2π≤x≤2π
Verified step by step guidance1
To find the inflection points, local maxima, and minima, first consider the function y = sin|x|. The absolute value affects the symmetry of the graph, making it even.
Identify critical points by finding where the derivative is zero or undefined. The derivative of y = sin|x| is y' = cos|x| * (x/|x|), which is undefined at x = 0.
Evaluate the second derivative to determine concavity. The second derivative will help identify intervals where the graph is concave up or concave down.
Analyze the graph visually to identify local maxima and minima. The peaks and troughs of the sine wave indicate these points, typically at x = ±π, ±2π.
Determine the open intervals of concavity. The graph is concave up where the second derivative is positive and concave down where it is negative. This typically alternates between intervals of π.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inflection Points
Inflection points occur where the concavity of a function changes from concave up to concave down or vice versa. To find these points, examine where the second derivative changes sign. For the function y = sin|x|, analyze the graph and derivatives to identify these transitions within the given interval.
Recommended video:
04:50
Critical Points
Local Maxima and Minima
Local maxima and minima are points where a function reaches a peak or a trough, respectively, within a certain interval. These can be found by setting the first derivative to zero and analyzing the sign changes. For y = sin|x|, observe the graph to determine where the function reaches its highest and lowest values within the specified range.
Recommended video:
07:09The First Derivative Test: Finding Local Extrema
Concavity and Differentiability
Concavity describes the direction a graph curves, either upwards (concave up) or downwards (concave down). Differentiability refers to the ability to compute a derivative at a point. For y = sin|x|, identify intervals of concavity by examining the second derivative and check differentiability by ensuring the function is smooth and continuous over the given domain.
Recommended video:
05:59Determining Concavity Given a Function
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 / (x² - 1)
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Textbook Question
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) = x − 6√(x − 1)
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Textbook Question
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.
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Textbook Question
Finding Functions from Derivatives
In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.
f'(x) = 2x − 1, P(0,0)
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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.
f(x) =√(x − 1), [1, 3]
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