   Chapter 3.6, Problem 22E

Chapter
Section
Textbook Problem

# 20–25 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 fx varies as c changes.

Explanation

1) Concept:

From the graph, identify the absolute minimum value, local maximum and local minimum points, decreasing, increasing of function and points of inflections

2) Given:

fx=xc2-x2

3) Calculation:

We have

fx=xc2-x2

There are two cases

i) c=0   and ii. c0

Here, when c=0 then fx consist only of origin and

When c0 then f(x) intersects x-axis in three points at x=-c, 0, c

f(x) is symmetric about origin means f(x) is an odd function

Differentiate f(x) with respect to x,

f'x=xc2-x2'+c2-x2x'

f'x=x-2x2c2-x2+c2-x21

f'x=-x2c2-x2+c2-x2

f'x=-x2+c2-x2c2-x2

f'x=-2x2+c2c2-x2

Take f'x=0

0=-2x2+c2c2-x2

-2x2+c2=0

x=±c2

Critical points are x=±c2

If c0,

fc2=c2c2-

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