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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Investigate the family of curves given by the equation f(x) = x4 + cx2 + x. Start by determining the transitional value of c at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. There is another transitional value of c at which the number of critical numbers changes. Try to discover it graphically. Then prove what you have discovered.

To determine

To find: The transitional value of c at which the number of inflection points changes for the curve function f(x)=x4+cx2+x .

Explanation

Given information:

The function is f(x)=x4+cx2+x .

Calculation:

Show the function as follows:

f(x)=x4+cx2+x (1)

Sketch the graph of the function f(x)=x4+cx2+x for different values of c as shown in Figure 1.

Differentiate Equation (1).

f(x)=4x3+2cx+1 (2)

Differentiate Equation (2).

f(x)=12x2+2c=2(6x2+c) (3)

If the value of c0 , then there is no inflection points.

Find the intersection point as shown below:

6x2+c=0x2=c6x=±c6

Refer to Figure 1.

For c=0 , the graph has one critical number at the absolute minimum around x=0.6 .

When c increases, the number of critical point does not change.

If c decreases from 0, the graph sprouts another local minimum to the right of origin between the points x=1 and x=2 .

At c=1.5 :

The absolute minimum value occurs at about x=1 and a horizontal tangent with no extreme at about x=0

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