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

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Chapter 4.5, Problem 70E
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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   5 x x 2

To determine

To graph: The sketch of the function y=ln5xx2 and label the relative extrema, points of inflection and asymptotes.

Explanation

Given Information:

The provided function is y=ln5xx2.

Graph:

Consider the function, y=ln5xx2

Evaluate the first derivative of the function, by applying quotient rule and constant multiple rule.

y′=ddx(ln5xx2)=x2ddxln5x−ln5xddxx2(x2)2=x2×55x−ln5x×2x(x)4=x−2xln5x(x)4

Further solve as,

y′=x(1−2ln5x)(x)4=1−2ln5x(x)3

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

1−2ln5x(x)3=01−2ln5x=02ln5x=1ln5x=12

Further solve as,

5x=e12x=15×1.6487x≈0.33

Hence, the critical number is x=0.33.

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

y″(x)=ddx[1−2ln5x(x)3]=(x)3ddx(1−2ln5x)−(1−2ln5x)ddx(x)3((x)3)2=(x)3×(−2×55x)−(1−2ln5x)×3x2x6=−2x2−3x2+6x2ln5xx6

Further solve as,

y″(x)=−5x2+6x2ln5xx6=x2(−5+6ln5x)x6=6ln5x−5x4

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

6ln5x−5x4=06ln5x−5=06ln5x=5ln5x=56

Further solve as,

5x=e56x=15×2.3009x≈0

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Elementary Geometry for College Students

FAMILY COMPOSITION In a four-child family, what is the expected number of boys? Assume that the probability of ...

Finite Mathematics for the Managerial, Life, and Social Sciences

Precalculus or Calculus In Exercises 3-6.decide whether the problem can be solved using precalculus or whether ...

Calculus

Using Initial Conditions In Exercises 17-22, solve thedifferential equation by the method of undeterminedcoeffi...

Multivariable Calculus

The equation of the plane formed by the two lines and is: 2(x − 3) − 4(y − 1) + (z − 5) = 0 (x − 3) + 2(y − 1...

Study Guide for Stewart's Multivariable Calculus, 8th

The average value of f(x) = 3x2 + 1 on the interval [2, 4] is: a) 29 b) 66 c) 58 d) 36

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

The J. R. Ryland Computer Company is considering a plant expansion to enable thecompany to begin production of ...

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

Refer to the data set in Table 2.18. a. Construct a scatter diagram to show the relationship between Year Found...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

The following data are costs (in cents) per ounce for nine different brands of sliced Swiss cheese (consumerrep...

Introduction To Statistics And Data Analysis

Given the cutting speed and the tool or workpiece diameter, determine the revolutions per minute in the followi...

Mathematics For Machine Technology

In Exercises 75-78, match the equation of the ellipse with its graph. (x2)249+(y+2)216=1 a. b. c. d.

College Algebra (MindTap Course List)

Define the concept of external validity and a threat to external validity.

Research Methods for the Behavioral Sciences (MindTap Course List)

Explain the distinction between naturalistic observation and participant observation, and describe the situatio...

Research Methods for the Behavioral Sciences (MindTap Course List)

Propulsion Neglecting air resistance and the weight of the propellant, determine the work done in propelling a ...

Calculus (MindTap Course List)

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