Textbook Question
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
g(x) = −x² − 6x − 9,−4 ≤ x < ∞
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 4.3.60a
Identifying Extrema
In Exercises 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = sec²x − 2tan x, −π/2 < x < π/2
Verified step by step guidance1
First, understand that local extrema occur where the derivative of the function is zero or undefined. Begin by finding the derivative of the function f(x) = sec²x − 2tan x.
To find the derivative, use the chain rule and trigonometric identities. The derivative of sec²x is 2sec²x tan x, and the derivative of -2tan x is -2sec²x. Therefore, f'(x) = 2sec²x tan x - 2sec²x.
Set the derivative f'(x) = 0 to find critical points. This gives the equation 2sec²x tan x - 2sec²x = 0. Simplify this to find tan x = 1.
Solve tan x = 1 within the interval -π/2 < x < π/2. The solution is x = π/4, as tan(π/4) = 1.
Evaluate the function f(x) at x = π/4 and check the endpoints of the interval to determine the local extrema. Compare these values to identify the local maximum and minimum.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Local Extrema
Local extrema refer to the points in a function where it reaches a local maximum or minimum. These are points where the function changes direction, and they can be found by analyzing the first derivative. A local maximum is where the function changes from increasing to decreasing, and a local minimum is where it changes from decreasing to increasing.
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Finding Extrema Graphically
First Derivative Test
The first derivative test is a method used to identify local extrema of a function. By taking the derivative of the function and finding its critical points (where the derivative is zero or undefined), we can determine where the function's slope changes sign. This change in sign indicates a local maximum or minimum, depending on the direction of the change.
Recommended video:
07:09The First Derivative Test: Finding Local Extrema
Trigonometric Derivatives
Understanding the derivatives of trigonometric functions is crucial for solving problems involving functions like sec²x and tan x. The derivative of sec²x is 2sec²x tan x, and the derivative of tan x is sec²x. These derivatives help in finding critical points and analyzing the behavior of the function within the given interval.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = (x − 2)²ᐟ³.
a. Does f′(2) exist?
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.
csc x cot x
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Textbook Question
Identifying Extrema
In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).
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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.
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.
(3/2)√x
Textbook Question
Identifying Extrema
In Exercises 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = √3cos x + sin x, 0 ≤ x ≤ 2π