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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.1.33
Chapter 4, Problem 4.1.33

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.


ƒ(x) = eˣ + e⁻ˣ

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To find the critical points of the function \( f(x) = e^x + e^{-x} \), we first need to find its derivative. The derivative of \( e^x \) is \( e^x \), and the derivative of \( e^{-x} \) is \( -e^{-x} \). Therefore, the derivative \( f'(x) \) is \( e^x - e^{-x} \).
Set the derivative equal to zero to find the critical points: \( e^x - e^{-x} = 0 \).
To solve \( e^x - e^{-x} = 0 \), add \( e^{-x} \) to both sides to get \( e^x = e^{-x} \).
Recognize that \( e^x = e^{-x} \) implies \( e^{2x} = 1 \) by multiplying both sides by \( e^x \).
Solve \( e^{2x} = 1 \) by taking the natural logarithm of both sides, which gives \( 2x = 0 \). Therefore, \( x = 0 \) is the critical point.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local maxima, minima, and points of inflection. To find critical points, one must first compute the derivative of the function and then solve for the values of x that satisfy the condition of the derivative being zero.
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Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides information about the function's slope at any given point. For the function ƒ(x) = eˣ + e⁻ˣ, the derivative can be calculated using the rules of differentiation, particularly the exponential function's properties.
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Derivatives

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. They are characterized by their rapid growth or decay and are crucial in various applications, including modeling population growth and radioactive decay. In the given function, eˣ and e⁻ˣ are examples of exponential functions, which will influence the behavior of the derivative and the location of critical points.
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Related Practice
Textbook Question

49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.


ƒ(x) = 1/3 x³ - 2x² - 5x + 2

Textbook Question

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


f(x) = x³ - (3/2)x² - 36x

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Textbook Question

Verify that the following functions satisfy the conditions of Theorem 4.9 on their domains. Then find the location and value of the absolute extrema guaranteed by the theorem.


f(x) = x√(3-x)

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Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (sec² Θ + sec Θ tan Θ)dΘ

Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ e (ln x - 1) / (x - 1)

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Textbook Question

Given a function f that is differentiable on its domain, write and explain the relationship between the differentials dx and dy.