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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 113-116, find all relative extrema of the function. Use a graphing utility to verify your result.

y = 3 ln  x x

To determine

To calculate: The all relative extrema of the function f(x)=3lnxx and verify the result using graphing utility.

Explanation

Given Information:

The provided function is f(x)=3lnxx.

Formula used:

Constant multiple rule is kg(x)=kg(x) where k is a constant.

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

f(x)=axa1

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

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, f(x)=3lnxx

Evaluate the derivative of the function by applying sum and difference rule.

f(x)=ddx(3lnxx)=ddx(3lnx)ddxx

Evaluate the derivative of the function by applying logarithm rule.

f(x)=ddx(3lnx)ddxx=31x1=3x1

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

3x1=03x=1x=3

Therefore, the critical number of the function is x=3

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