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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

Investigate the family of curves given by

f ( x ) = x 4 + x 3 + c x 2

In particular you should determine the transitional value of c at which the number of critical numbers changes and the transitional value at which the number of inflection points changes. Illustrate the various possible shapes with graphs.

To determine

The transitional value of c at which the number of critical number changes.

Explanation

1) Concept:

 Definition of a critical number: A critical number of a function f   is a number c in the domain of f  Such that either  f'c=0 or f'c does not exist.

To get the critical numbers, differentiate f(x) and find the values of x where f'x=0, and f'x doesn’t exists.

2) Given:

 fx=x4+x3+cx2

3) Calculation:

Consider the given family of curve,

fx=x4+x3+cx2

Using the power rule of differentiation, differentiate f(x) with respect to x,

f'x=d(x4+x3+cx2)dx

f'x=dx4dx+d(x3)dx+dcx2dx

f'x=4x3+3x2+2cx

f'x exists at all numbers because it is a polynomial function.

 To find the critical number solve for f'x=0

Hence,

 f'x=4x3+3x2+2cx=0

Factor out x  from the equation,

f'x=x4x2+3x+2c=0x=0 or x=-34

If 4x2+3x+2c=0

Using the determinant method to solve the quadratic equation, find the roots of the polynomial 4x2+3x+2c=0

 x=-3±-3±32-4·4·2c2·4

By simplifying,

x=-3±9-32c8

The discriminant 9-32c  is negative when 9-32c<0 i.e.9<32c=932<c

If c=932  then discriminant of the quadratic equation is 0 then the quadratic equation has repeated roots.

At c=932 there are two critical points 0 &-38

If 0<c<9 32  there are three critical points 0, -3+9-2c8 and -3-9-32c8 

If c = 0 for  f'x=0 then f'x=4x3+3x2=0

Factor out x2from 4x3+3x2=0 it becomes

x24x+3=0

 x24x+3=0 iff x2=0 or 4x+3=0  i.e

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