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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
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 3 + c x

To determine

To describe:

How the graph of fx varies as c varies.

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=x3+cx

3) Calculation:

fx=x3+cx

Differentiate f(x) with respect to x,

f'x=3x2+c

Again Differentiate with respect to x, we get

f''x=6x

 f''x>0 For x>0 then f(x) is concave up for (0, )

 f''x<0 For x<0 then f(x) is concave down for 0(-, 0)

Therefore, inflection point is x = 0

Consider f'x=3x2+c

There are three possibilities for c,

i. c>0,

 f'x>0, for all x

f(x) is increasing function

The f(x) has no maximum and minimum value for c>0

ii. c=0,

 f'x>0, for all x

f(x) is increasing function

The f(x) has no maximum and minimum value for c=0

iii. c<0

f'x=3x2+c

Substitute f'=0

0=3x2+c

x=±-c3

 f'x0-c3<x<--c3

That is the f(x) is decreasing on --c3, -c3

f(x) is increasing on -,--c3-c3, 

So the critical points are x=--c3, -

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