   Chapter 4.6, Problem 40E ### 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

# (a) Investigate the family of polynomials given by the equation f(x) = 2x3 + cx2 + 2x. For what values of c does the curve have maximum and minimum points?(b) Show that the minimum and maximum points of every curve in the family lie on the curve y = x − x3. Illustrate by graphing this curve and several members of the family.

(a)

To determine

To find: The minimum and maximum points of the curve for different values of c.

Explanation

Given:

The function is f(x)=2x3+cx2+2x

Calculation:

Show the Equation as follows:

f(x)=2x3+cx2+2x (1)

Sketch the graph for the function f(x)=2x3+cx2+2x for different values of c as shown in Figure 1.

Differentiate Equation (1).

f(x)=6x2+2cx+2 (2)

Modify Equation (2).

f(x)=2(3x2+cx+1)=6x2+2cx+2

Apply the quadratic formula to solve above equation as follows:

x=b±b24ac2a (3)

Substitute c for b, 3 for a, and 1 for c in Equation (3).

x=c±c24(3)(1)2(3)=c±c2126

Thus, f has critical points

(b)

To determine

To show: The minimum and maximum points of the curves.

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