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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Chapter
Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

(a) Find the intervals of increase or decrease.

(b) Find the local maximum and minimum values.

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

(d) Use the information from parts (a)–(c) to sketch the graph. Check your work with a graphing device if you have one.

f ( x ) = 1 2 x 4 4 x 2 + 3

(a)

To determine

To find: On which intervals the function f(x)=12x44x2+3 is increasing or decreasing.

Explanation

Given:

The function is, f(x)=12x44x2+3 .

Calculation:

Obtain the derivative of f(x) .

f(x)=ddx(12x44x2+3)=12(4x3)4(2x)+0=2x38x

Set f(x)=0 and solve for x.

2x38x=02x(x24)2x(x+2)(x2)2x=0,x+2=0,x2=0

Thus, the critical numbers are x=0 , x=2 and x=2 . In the interval notation (,2)(2,0)(0,2)(2,)

(b)

To determine

To find: The local maximum and minimum values of the function f(x)=12x44x2+3 .

(c)

To determine

To find: The intervals of concavity and the inflection points of the function f(x)=12x44x2+3 .

(d)

To determine

To sketch: The graph of the function using the information obtained from part (a) to part (c).

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