   Chapter 4.6, Problem 30E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of c at which the basic shape of the curve changes. f ( x ) = x c 2 − x 2

To determine

To describe: How the graph of f varies as c varies.

Explanation

Given information:

The function is f(x)=xc2x2 .

Calculation:

Show the function as follows:

f(x)=xc2x2 (1)

Differentiate both sides of the Equation (1).

f(x)=x12(c2x2)121(2x)+(c2x2)12(1)=x2c2x2+c2x2=x2+c2x2c2x2=c22x2c2x2

Differentiate both sides of the Equation.

f(x)=c2x2(4x)(c22x2)12(c2x2)12(2x)(c2x2)2=4xc2x2+x(c22x2)c2x2(c2x2)=4x(c2x2)+c2x2x3c2x2(c2x2)=4c2x+4x3+c2x2x3(c2x2)32

=2x33c2x(c2x2)32

Substitute x for x in Equation (1).

f(x)=xc2(x)2=xc2x2=f(x)

Hence, the function is an odd function. So the graph is symmetric about the origin.

Sketch the function for different values of c as shown in Figure 1.

Consider the value c>0

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 