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
Not the one you use?Change textbookSelect a different textbook
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 4.3.55a
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
Verified step by step guidance1
To find the local extrema of the function \( f(x) = \sqrt{3} \cos x + \sin x \) on the interval \( 0 \leq x \leq 2\pi \), start by finding the derivative of the function. The derivative \( f'(x) \) is given by \( f'(x) = -\sqrt{3} \sin x + \cos x \).
Set the derivative equal to zero to find the critical points: \( -\sqrt{3} \sin x + \cos x = 0 \). Rearrange this equation to \( \cos x = \sqrt{3} \sin x \).
Divide both sides by \( \cos x \) (assuming \( \cos x \neq 0 \)) to get \( 1 = \sqrt{3} \tan x \), which simplifies to \( \tan x = \frac{1}{\sqrt{3}} \).
Solve \( \tan x = \frac{1}{\sqrt{3}} \) for \( x \) within the interval \( 0 \leq x \leq 2\pi \). The solutions are \( x = \frac{\pi}{6} \) and \( x = \frac{7\pi}{6} \).
Evaluate the function \( f(x) \) at the critical points \( x = \frac{\pi}{6} \), \( x = \frac{7\pi}{6} \), and the endpoints \( x = 0 \) and \( x = 2\pi \) to determine the local extrema. Compare these values to identify the local maxima and minima.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7mWas this helpful?
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 value. These are points where the function changes direction, and they can be found by analyzing the first derivative. A local maximum occurs when the derivative changes from positive to negative, while a local minimum occurs when it changes from negative to positive.
Recommended video:
05:58
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. Analyzing the sign of the derivative before and after these points helps identify whether they are local maxima or minima.
Recommended video:
07:09The First Derivative Test: Finding Local Extrema
Trigonometric Functions
Trigonometric functions, such as sine and cosine, are periodic functions that describe oscillatory behavior. In the context of finding extrema, understanding their periodicity and behavior over specific intervals, like 0 to 2π, is crucial. These functions have known maximum and minimum values, which can help in determining the extrema of more complex functions involving them.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
51. Frictionless cart A small frictionless cart, attached to the wall by a spring, is pulled 10 cm from its rest position and released at time t = 0 to roll back and forth for 4 sec. Its position at time t is s = 10 cos πt.
a. What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then?
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) = sec²x − 2tan x, −π/2 < x < π/2
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
1
views
Textbook Question
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
a. What are the critical points of f?
f′(x) = (x − 1)²(x + 2)
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.
f(t) = t³ − 3t², −∞ < t ≤ 3