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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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Find all relative extrema of the function f ( x ) = x 2 ln x .

To determine

To calculate: The relative extrema of the function f(x)=x2ln x.

Explanation

Given Information:

The provided function is f(x)=x2ln x.

Formula used:

Rules for derivatives:

Sum and difference rule of derivative:

(f(x)±g(x))=f(x)±g(x)

Derivative of function f(x)=ln x is,

ddxlnx=1x

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

Derivative of function y=xn using power rule is dydx=nxn1.

Calculation:

Consider the function f(x)=x2ln x.

First, evaluate the first derivative of the function f(x)=x2ln x.

dydx=ddx(x2ln x)

Apply the difference rule,

ddx(x2ln x)=ddx(x)ddx(2lnx)=12×1x=12x

Hence, the first derivative of the function is dydx=12x.

Now, set the derivative equal to 0.

12x=02x=1x=2

Hence, the only critical numbers is x=2

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