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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.3.25a
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 guidance
1
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.

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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 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.
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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.
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The 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.
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Determining Where a Function is Increasing & Decreasing
Related Practice
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.

−πsin πx

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

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

sec²x

2
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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

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


f(x) = (x + 1)², −∞ < x ≤ 0

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.

∫ sec θ/3 tan θ/3 dθ