Explanation Given: The function is f ( x ) = x 2 x − 2 . Calculation: For any function, the critical numbers are the solutions of f ′ ( x ) = 0 Obtain f ′ ( x ) is as follows. f ′ ( x ) = d d x ( x 2 x − 2 ) = ( x − 2 ) d d x ( x 2 ) − x 2 d d x ( x − 2 ) ( x − 2 ) 2 = ( x − 2 ) 2 x − x 2 ( x − 2 ) 2 = 2 x 2 − 4 x − x 2 ( x − 2 ) 2 = x 2 − 4 x ( x − 2 ) 2 Thus, the value is f ′ ( x ) = x 2 − 4 x ( x − 2 ) 2 . Equate f ′ ( x ) = 0 x 2 − 4 x ( x − 2 ) 2 = 0 x 2 − 4 x = 0 x ( x − 4 ) = 0 x = 0 and x = 4 Thus, x = 0 and x = 4 are the partition numbers. There is also a partition number where the function f ( x ) = x 2 x − 2 is not defined. The function f ( x ) = x 2 x − 2 is not defined when the denominator is 0. So, x = 2 is also a partition number. Therefore, the partition numbers are x = 0 , x = 2 and x = 4 . The given function f ( x ) = x 2 x − 2 is defined on ( − ∞ , 2 ) ∪ ( 2 , ∞ ) . So, the domain is ( − ∞ , 2 ) ∪ ( 2 , ∞ ) . The partition numbers x = 0 and x = 4 lies in the domain but x = 2 does not lie in the domain. Therefore, the critical numbers of f ( x ) are x = 0 and x = 4 . The function f ( x ) is increasing when its derivative f ′ ( x ) > 0 and decreasing when f ′ ( x ) < 0 . Since x = 0 , x = 2 and x = 4 are the partition number, the domain can be divided into four intervals as. ( − ∞ , 0 ) , ( 0 , 2 ) , ( 2 , 4 ) and ( 4 , ∞ ) Let x = − 1 be a test number in the interval ( − ∞ , 0 ) . Substitute x = − 1 in f ′ ( x ) = x 2 − 4 x ( x − 2 ) 2 . f ′ ( − 1 ) = ( − 1 ) 2 − 4 ( − 1 ) ( − 1 − 2 ) 2 = 1 + 4 9 = 5 9 > 0 Since f ′ ( x ) > 0 , the function f ( x ) is increasing in the interval ( − ∞ , 0 )

College Mathematics for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134674148

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Question

Chapter 11.1, Problem 89E

To determine

To find: The critical numbers of the function, the intervals where the function is increasing or decreasing and the local extrema.

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Chapter 11.1, Problem 88E

Chapter 11.1, Problem 90E

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Chapter 11 Solutions

College Mathematics for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 11.1 - Matched Problem 1 Repeat Example 1 for f(x) = x2 –...Ch. 11.1 - Matches Problem 2 Find the critical numbers of f,...Ch. 11.1 - Matched Problem 3 Find the critical numbers of f,...Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 7MP Ch. 11.1 - Prob. 8MP Ch. 11.1 - Prob. 9MP Ch. 11.1 - Skills Warm-up Exercises In Problems 1-8, inspect...

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