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
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.3.25a
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
f(r) = 3r³ + 16r
Verified step by step guidance1
First, find the derivative of the function f(r) = 3r³ + 16r. The derivative, f'(r), will help us determine where the function is increasing or decreasing.
Calculate the derivative: f'(r) = d/dr (3r³ + 16r). Use the power rule for differentiation, which states that d/dr (r^n) = n*r^(n-1).
Apply the power rule: f'(r) = 3 * 3r² + 16 * 1 = 9r² + 16.
Set the derivative f'(r) = 9r² + 16 equal to zero to find critical points, which are potential points where the function changes from increasing to decreasing or vice versa.
Solve the equation 9r² + 16 = 0 for r. These critical points will help determine the intervals of increase and decrease. Analyze the sign of f'(r) around these points to identify the intervals where the function is increasing or decreasing.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5mKey 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 zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. For the function f(r) = 3r³ + 16r, finding the derivative and setting it to zero will help identify these critical points.
Recommended video:
04:50
Critical Points
First Derivative Test
The First Derivative Test is used to determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign of the derivative before and after each critical point, we can ascertain the intervals where the function is increasing or decreasing. This test is crucial for understanding the behavior of f(r) = 3r³ + 16r around its critical points.
Recommended video:
07:09The First Derivative Test: Finding Local Extrema
Increasing and Decreasing Intervals
A function is increasing on an interval if its derivative is positive throughout that interval, and decreasing if its derivative is negative. By applying the First Derivative Test to f(r) = 3r³ + 16r, we can identify these intervals, which are key to understanding the overall shape and behavior of the function.
Recommended video:
07:32Determining Where a Function is Increasing & Decreasing
Related Practice
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) = sin 2x, 0 ≤ x ≤ π
Textbook Question
Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫ 1/( r + 5)²dr
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(x − 1)
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) = sin x − cos x,0 ≤ x ≤ 2π
Textbook Question
25. Paper folding A rectangular sheet of 8.5-in.-by-11-in. paper is placed on a flat surface. One of the corners is placed on the opposite longer edge, as shown in the figure, and held there as the paper is smoothed flat. The problem is to make the length of the crease as small as possible. Call the length L. Try it with paper.
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a. Show that L^2=2x^3/(2x-8.5).