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

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Chapter 4.5, Problem 67E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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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 65-68, find all relative extrema of the function. Use a graphing utility to verify your result. See Example 6.

y = ln ( 1 − 4 x − x 2 )

To determine

To calculate: The relative extrema of the function y=ln(1−4x−x2).

Explanation

Given Information:

The provided function is y=ln(1−4x−x2).

Formula used:

Rules for derivatives:

Sum rule of derivative:

(f(x)+g(x))'=f'(x)+g'(x)

Derivative of function f(x)=ln x is,

ddxlnx=1x

Constant multiple rule is f'(cx)=cf(x) where c is constant.

Derivative of function y=xn using power rule is dydx=nxn−1.

If f(x) and g(x) are two functions then quotient rule of derivative of functions is,

ddx[f(x)g(x)]=g(x)⋅f′(x)−f(x)⋅g′(x)[g(x)]2

Calculation:

Consider the function y=ln(1−4x−x2).

First, evaluate the first derivative of the function y=ln(1−4x−x2).

dydx=ddxln(1−4x−x2)

Apply the chain rule,

ddxln(1−4x−x2)=ddxln(1−4x−x2)×ddx(1−4x−x2)=1(1−4x−x2)[ddx1−ddx4x−ddx(x2)]=1(1−4x−x2)[0−4−2x]=−4−2x(1−4x−x2)

Hence, the first derivative of the function is dydx=−4−2x(1−4x−x2).

Now, set the derivative equal to 0.

−4−2x(1−4x−x2)=0−4−2x=0−2x=4x=−2

Hence, the only critical number is x=−2

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Calculus

Volume and Surface AreaShow that a rectangular box of given volume and minimum surface area is a cube.

Multivariable Calculus

The nth term in the Taylor series centered at 1 for f(x) = x−2 is: (−1)n + 1(n + 1)!(x − 1)n (−1)nn!(x − 1)n (−...

Study Guide for Stewart's Multivariable Calculus, 8th

Evaluate dx using the Fundamental Theorem of Calculus. a) 7 b) 9 c) d) 18

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

In addition to the key words, you should also be able to define each of the following terms: Positive relations...

Research Methods for the Behavioral Sciences (MindTap Course List)

Applications Dimensions of a white board The area of a SMART Board interactive white board is 2880 square inche...

College Algebra (MindTap Course List)

The Wall Street Journal (February 15, 1972) reported that General Electric was sued in Texas for sex discrimina...

Introduction To Statistics And Data Analysis

Describe the four methods for administering a survey (mail, phone, Internet, and in person) and explain the str...

Research Methods for the Behavioral Sciences (MindTap Course List)

Find dimension y. All dimensions are in inches

Mathematics For Machine Technology

Community Bank would like to increase the number of customers who use payroll direct deposit. Management is con...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

A drug store chain has stores in four cities: 38 stores in Indianapolis. 45 in Louisville, 80 in St. Louis, and...

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

Using the Squeeze Theorem In Exercises 93-96, use a graphing utility to graph the given function and the equati...

Calculus (MindTap Course List)

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