Textbook Question
Identifying Extrema
In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.
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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 4.5
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
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y = √𝓍² ― 1
Verified step by step guidance1
Identify the natural domain of the function y = √(x² - 1). The expression under the square root, x² - 1, must be greater than or equal to zero for y to be real. Solve the inequality x² - 1 ≥ 0 to find the domain.
Solve the inequality x² - 1 ≥ 0. This can be rewritten as x² ≥ 1, which implies x ≤ -1 or x ≥ 1. Therefore, the natural domain of the function is x ∈ (-∞, -1] ∪ [1, ∞).
Find the critical points by taking the derivative of the function. The derivative of y = √(x² - 1) is y' = (1/2)(x² - 1)^(-1/2) * 2x = x / √(x² - 1). Set y' = 0 to find critical points.
Solve the equation x / √(x² - 1) = 0. This implies x = 0. However, x = 0 is not in the domain of the function, so there are no critical points within the domain.
Evaluate the function at the endpoints of the domain, x = -1 and x = 1, to find the extreme values. Calculate y(-1) and y(1) to determine the absolute minimum and maximum values of the function over its natural domain.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Extreme Values
Extreme values refer to the maximum and minimum values of a function within a given domain. Absolute extreme values are the highest and lowest points over the entire domain, while local extreme values are the highest or lowest points within a specific interval. Identifying these values often involves analyzing the function's critical points and endpoints.
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Critical Points
Critical points are values in the domain of a function where the derivative is either zero or undefined. These points are essential for finding local extrema, as they indicate where the function's slope changes, potentially leading to local maxima or minima. To find critical points, one typically takes the derivative of the function and solves for when it equals zero.
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Natural Domain
The natural domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function y = √(x² - 1), the natural domain is determined by ensuring the expression under the square root is non-negative, leading to the condition x² - 1 ≥ 0. This results in the domain being x ≤ -1 or x ≥ 1.
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Related Practice
Textbook Question
Identifying Extrema
In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.
178
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Textbook Question
Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
f(x) = (2/3)x − 5, −2 ≤ x ≤ 3
210
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Textbook Question
54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)
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Textbook Question
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.
2. y=x^4/4-2x^2+4
223
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Textbook Question
Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
f(t) = 2 − |t|, −1 ≤ t ≤ 3
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