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 4.3.18
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
To determine where the function is increasing or decreasing, observe the graph of the function. The function is increasing where the slope of the graph is positive and decreasing where the slope is negative.
Identify the intervals where the function is increasing. From the graph, the function appears to be increasing on the interval (-2, 1).
Identify the intervals where the function is decreasing. From the graph, the function appears to be decreasing on the intervals (-∞, -2) and (1, ∞).
To find local extrema, look for points where the function changes from increasing to decreasing or vice versa. The graph shows a local maximum at x = -2 and a local minimum at x = 1.
Check for absolute extrema by comparing the values of the function at critical points and endpoints. The graph suggests an absolute maximum at x = -2 and an absolute minimum at x = 1.
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Key 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 numbers x1 and x2 within that interval, x1 < x2 implies f(x1) < f(x2). Conversely, it is decreasing if x1 < x2 implies f(x1) > f(x2). Identifying these intervals involves analyzing the function's derivative: where it is positive, the function is increasing, and where it is negative, the function is 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 minimum or maximum value within a neighborhood, while absolute extrema are the highest or lowest points over the entire domain. To find these, examine critical points where the derivative is zero or undefined, and evaluate the function's behavior at these points and endpoints of the domain.
Recommended video:
05:58Finding Extrema Graphically
Critical Points
Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. To determine the nature of these points, use the first or second derivative test. The first derivative test involves checking sign changes around the critical points, while the second derivative test uses concavity to identify extrema.
Recommended video:
04:50Critical Points
Related Practice
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)
Textbook Question
26. Constructing cylinders Compare the answers to the following two construction problems.
a. A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?
b. The same sheet is to be revolved about one of the sides of length y to sweep out the cylinder as shown in part (b) of the figure. What values of x and y give the largest volume?
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Textbook Question
107. Marginal cost The accompanying graph shows the hypothetical cost c=f(x) of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing?
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Textbook Question
Theory and Examples
In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.
y = x¹¹ + x³ + x − 5
Textbook Question
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
g(t) = √t + √(1 + t) − 4, (0, ∞)