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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 4, Problem 115RE
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Ron Larson
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding Relative Extrema In Exercises 113-116, find all relative extrema of the function. Use a graphing utility to verify your result.

y = ln ( x 2 + 2 x + 7 )

To determine

To calculate: The all relative extrema of the function y=ln(x2+2x+7) and verify the result using graphing utility.

Explanation

Given Information:

The provided function is y=ln(x2+2x+7).

Formula used:

If the function is f(x)=xa, then the power rule of the derivative of the function is given by.

f(x)=axa−1

If the function is f(x)=[lnt], then the chain rule for logarithmic function is given by.

f′(x)=1t.dtdx

Calculation:

Consider the function, y=ln(x2+2x+7)

Evaluate the derivative of the function by applying logarithm rule.

y′=ddxln(x2+2x+7)=ddxln(x2+2x+7)×ddx(x2+2x+7)=1(x2+2x+7)×(2x+2)=2(x+1)x2+2x+7

Set the derivative equal to zero to compute the value of x.

2(x+1)x2+2x+7=02(x+1)=0x+1=0x=−1

Therefore, the critical number of the function is x=−1.

Evaluate the result of substitution the values from each region in the derivative of the function.

Consider the interval, −∞<x<−1

Test the region by substituting the value from the region in the derivative function.

y′(−2)=2(−2+1)(−2)2+2(−2)+7=2×−14−4+7=−27=−0.286

The test for the value from this region is negative, therefore the function is decreasing.

Consider the point, x=−1

Test the region by substituting the value from the region in the derivative function

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Sect-4.1 P-1CPSect-4.1 P-2CPSect-4.1 P-3CPSect-4.1 P-4CPSect-4.1 P-5CPSect-4.1 P-1SWUSect-4.1 P-2SWUSect-4.1 P-3SWUSect-4.1 P-4SWUSect-4.1 P-5SWU
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Contemporary Mathematics for Business & Consumers

Work An automobile is pushed down a level street by exerting a force of 85 pounds at an angle of 15 with the ho...

Trigonometry (MindTap Course List)

A builder sells an 1800-ft2 home for $171,000. What would be the price of a 2400-ft2 home of similar quality? A...

Elementary Technical Mathematics

In Exercises 7 to 12, simplify by using the FOIL method of multiplication. x+3x+4

Elementary Geometry For College Students, 7e

Testing for Continuity In Exercises 75-82, describe the interval(s) on which the function is continuous. f(x)=3...

Calculus: Early Transcendental Functions (MindTap Course List)

Comparing y and dy In Exercises 95 and 96, use the information to find and compare y and dy. Function x-Value...

Calculus of a Single Variable

MINIMIZING CITY WATER COSTS The water-supply manager for a Midwestern city needs to supply the city with at lea...

Finite Mathematics for the Managerial, Life, and Social Sciences

In Exercises 21 to 24, the triangles named can be proved congruent. Considering the congruent pairs marked, nam...

Elementary Geometry for College Students

For , which vector, a, b, c, or d, best represents F(1, 1)? a b c d

Study Guide for Stewart's Multivariable Calculus, 8th

Marginal Revenue A pharmaceutical corporation has two plants that produce the same over-the-counter medicine. I...

Calculus

True or False: n=1(1)nnn+3 converges.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Volume of a Region Bounded by Two Surfaces In Exercises 35-40, set up a double integral to find the volume of t...

Calculus (MindTap Course List)

Sketching a Vector Field In Exercises 914, find F and sketch several representative vectors in the vector fie...

Multivariable Calculus

Define, compare, and contrast the experimental, nonexperimental, and quasi-experimental research strategies, an...

Research Methods for the Behavioral Sciences (MindTap Course List)

A sample of 426 U.S. adults age 50 and older were asked how important a variety of issues were in choosing whom...

Statistics for Business & Economics, Revised (MindTap Course List)

Decompose each fraction into partial fractions. 5x2+9x+3xx+12

College Algebra (MindTap Course List)

A CBS News/New York Times survey found that 97% of Americans believe that textingwhile driving should be outlaw...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Refer to the previous exercise. Suppose now that for a probability question, 100 answers are submitted, of whic...

Introduction To Statistics And Data Analysis

For each exercise, functions of two angles are given. Which of the functions of the two angles is greater? Do n...

Mathematics For Machine Technology

Describe the purpose of a phase in a single-case design and explain how patterns within a phase are defined, in...

Research Methods for the Behavioral Sciences (MindTap Course List)

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