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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 x x

To determine

To calculate: The relative extrema of the function y=ln xx and verify the result by graphing utility tool.

Explanation

Given Information:

The provided function is y=ln xx.

Formula used:

Sum rule of derivative:

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

Derivative of logarithm function f(x)=ln x is,

ddxlnx=1x

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

The simple power rule of derivation,

dxndx=nxn1

Calculation:

Consider the function y=ln xx.

First, evaluate the first derivative of the function y=ln xx.

Use the sum rule of derivative, differentiate the provided function.

dydx=ddx(ln xx)=ddx(ln x)ddx(x)=1x1

Use the derivation of logarithm function and simple power rule of derivation,

dydx=1x1

Hence, the first derivative of the function is dydx=1x1.

Now, equate the first derivative equal to 0.

1x1=01x=1x=1

Therefore, the critical value is 1.

Use the sum rule of derivative, differentiate the first derivative 1x1.

ddx(1x1)=ddx(1x)ddx(1)=ddx(x1)0=(1)x11=x2

Therefore, the second derivative is x2

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