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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

9-14

(a) Find the intervals on which f is increasing or decreasing.

(b) Find the local maximum and minimum values of f.

(c) Find the intervals of concavity and the inflection points.

f ( x ) = x x 2 + 1

To determine

(a)

To find:

The intervals on which f  is increasing or decreasing

Explanation

1) Concept:

The derivative of a function can be used to determine whether the function is increasing or decreasing on any intervals in its domain.

If f'x>0  over an interval I,  then fx is increasing on I.

If f'x<0  over an interval I, then fx is decreasing on I.

2) Calculation:

fx=xx2+1

f'x=x2+1 1-x(2x)(x2+1)2=1-x2(x2+1)2

f'x= 1-x2(x2+1)2 = (1+x)(1-x)(x2+1)2

Set f'x=0

x+1x-1

To determine

(b)

To find:

Local maximum and local minimum value of f(x)

To determine

(c)

To find:

Intervals of concavity and inflection points

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