   Chapter 3.6, Problem 23E

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 ) = c x 1 + 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=cx1+c2x2

3) Calculation:

Here,

fx=cx1+c2x2

We draw the graph of f(x)

There are two cases c>0  and c<0

Here when c=0 then f(x) consist only on origin and

When c>0, c<0 then f(x) intersects x-axis at origin

f(x) has horizontal asymptote because limn±cx1+c2x2=±

Now, find out maximum and minimum points,

Differentiate f(x) on both sides with respect to x, we get

fx=cx1+c2x2

f'x=1+c2x2cx'-cx1+c2x2'1+c2x22

f'x=1+c2x2c-cx2c2x1+c2x22

f'x=c2+c3x2-2c3x21+c2x22

f'x=c2-c3x21+c2x22

Take f'x=0

0=c2-c3x21+c2x22

c2

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