   Chapter 11.1, Problem 40E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In Problems 39-42, find the relative maxima and relative minima, and sketch the graph with a graphing calculator to check your results. y = x 2 ln x

To determine

To calculate: The value of relative maxima and relative minima for the function y=x2lnx and sketch the graph using graphing calculator to check.

Explanation

Given Information:

The function is y=x2lnx.

Formula Used:

Product rule,

ddx(uv)=uv+uv, where u and v are two separate functions.

Derivative of logarithmic function,

ddx(lnx)=1x

Power rule of differentiation,

ddx(xn)=nxn1

Chain rule of differentiation,

ddxf(g(x))=f(g(x))g(x)

Calculation:

Consider the provided function y=x2lnx.

First calculate the derivative of the function.

Recall the product rule of derivatives,

ddx(uv)=uv+uv

Apply the product rule ton given function to get,

y=ddx(x2lnx)=ddxx2ln(x)+x2ddx(lnx)

Recall, the chain rule of derivatives.

By chain rule,

ddxf(g(x))=fg(x)g(x)

Apply the chain rule to get,

y=ddxx2ln(x)+x2ddx(lnx)=ddxx2ln(x)+x2ddx(lnx)ddxx

Recall, the property of derivative,

Properties of derivative,

ddx(lnx)=1x

Also, recall the power rule,

ddx(xn)=nxn1

Apply the properties on the given function to get,

y=ddxx2ln(x)+x2ddx(lnx)ddxx=2x21lnx+x21x1=2xlnx+x=2xlnx+x

To find the relative maxima and relative minima put the derivative of the function equal to zero to get the value of x,

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