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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 = x ln 5 x

To determine

To calculate: The all relative extrema of the function f(x)=xln5x and verify the result using graphing utility.

Explanation

Given Information:

The provided function is f(x)=xln5x.

Formula used:

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

f(x)=axa1

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)=xln5x

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

f(x)=ddx(xln5x)=ddxxddx(ln5x)

Evaluate the derivative of the function by applying logarithm rule.

f(x)=ddxxddx(ln5+lnx)=1ddx(ln5)ddxlnx=101x=11x

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

11x=01x=1x=1

Therefore, the critical number of the function is x=1.

Evaluate the result of substitution the values from each region in the derivative of the function.

Consider the interval, 0<x<1

Test the region by substituting the value from the region in the derivative function.

f(0.5)=110.5=12=1

The test for the value from this region is negative, therefore the function is decreasing

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