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

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Chapter 4.5, Problem 72E
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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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Analyzing a Graph In Exercises 69-72, analyze and sketch the graph of the function. Label any relative extrema, points of inflection, and asymptotes.

y = ( ln x ) 2

To determine

To graph: The sketch of the function y=(lnx)2 and label the relative extrema, points of inflection and asymptotes.

Explanation

Given Information:

The provided function is y=(lnx)2.

Graph:

Consider the function, y=(lnx)2

Evaluate the first derivative of the function, by applying chain rule.

y′=ddx(lnx)2=2lnx(1x)=2lnxx

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

2lnxx=02lnx=0lnx=0x=e0

Further solve as,

x=1

Hence, the critical number is x=1.

Now, evaluate the second derivative of the function by applying quotient rule.

y″(x)=ddx[2lnxx]=xddx2lnx−2lnxddxxx2=2x×1x−2lnxx2=2−2lnxx2

Set the second derivative equal to zero, to determine the value of x.

2−2lnxx2=02−2lnx=02lnx=2lnx=1

Further solve as,

x=e1x=2.718

Substitute x=2.718 in the function.

y(2.718)=(ln2.718)2=(0.99989)2≈1

Therefore, the point of inflection is (2

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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
Sect-4.1 P-6SWUSect-4.1 P-7SWUSect-4.1 P-8SWUSect-4.1 P-9SWUSect-4.1 P-10SWUSect-4.1 P-11SWUSect-4.1 P-12SWUSect-4.1 P-13SWUSect-4.1 P-14SWUSect-4.1 P-15SWUSect-4.1 P-16SWUSect-4.1 P-17SWUSect-4.1 P-18SWUSect-4.1 P-1ESect-4.1 P-2ESect-4.1 P-3ESect-4.1 P-4ESect-4.1 P-5ESect-4.1 P-6ESect-4.1 P-7ESect-4.1 P-8ESect-4.1 P-9ESect-4.1 P-10ESect-4.1 P-11ESect-4.1 P-12ESect-4.1 P-13ESect-4.1 P-14ESect-4.1 P-15ESect-4.1 P-16ESect-4.1 P-17ESect-4.1 P-18ESect-4.1 P-19ESect-4.1 P-20ESect-4.1 P-21ESect-4.1 P-22ESect-4.1 P-23ESect-4.1 P-24ESect-4.1 P-25ESect-4.1 P-26ESect-4.1 P-27ESect-4.2 P-1CPSect-4.2 P-2CPSect-4.2 P-3CPSect-4.2 P-4CPSect-4.2 P-5CPSect-4.2 P-1SWUSect-4.2 P-2SWUSect-4.2 P-3SWUSect-4.2 P-4SWUSect-4.2 P-5SWUSect-4.2 P-6SWUSect-4.2 P-7SWUSect-4.2 P-8SWUSect-4.2 P-9SWUSect-4.2 P-10SWUSect-4.2 P-11SWUSect-4.2 P-12SWUSect-4.2 P-1ESect-4.2 P-2ESect-4.2 P-3ESect-4.2 P-4ESect-4.2 P-5ESect-4.2 P-6ESect-4.2 P-7ESect-4.2 P-8ESect-4.2 P-9ESect-4.2 P-10ESect-4.2 P-11ESect-4.2 P-12ESect-4.2 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