   Chapter 3.6, Problem 28E

Chapter
Section
Textbook Problem

# (a) Investigate the family of polynomials given by the equation f ( x ) = 2 x 3 + c x 2 + 2 x . For what values of c does the curve have maximum and minimum points?(b) Show that the minimum and maximum points of everycurve in the family lie on the curve y = x − x 3 . Illustrate by graphing this curve and several members of the family.

To determine

a)

To find:

The values of c which have maximum and minimum points

Explanation

1) Given:

fx=2x3+cx2+2x

2) Calculation:

Here,fx=2x3+cx2+2x

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

f'x=6x2+2cx+2

Substitute f'x=0

0=6x2+2cx+2

3x2+cx+1=0

So solve this equation by formula method,

x=-c±c2-126

Therefore, f has critical point ⇔

To determine

b)

To show:

The minimum and maximum points of the curve that lie on the curve y=x-x3

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