Explanation For any function the critical numbers are the solutions of f ′ ( x ) = 0 Evaluate f ′ ( x ) as follows. f ′ ( x ) = d d x ( x 4 − 18 x 2 ) = 4 x 3 − 18 ( 2 x ) = 4 x 3 − 36 x Thus, f ′ ( x ) = 4 x 3 − 36 x . Equate f ′ ( x ) = 0 4 x 3 − 36 x = 0 x 3 − 9 x = 0 x ( x 2 − 9 ) = 0 x ( x + 3 ) ( x − 3 ) = 0 Thus, x = 0 , x = − 3 and x = 3 are the partition numbers. The given function is a polynomial function and it is defined for all x, so the domain of the function is ( − ∞ , ∞ ) . Since all the partition numbers lies in the domain, they are called as critical numbers. The function f ( x ) is increasing when its derivative f ′ ( x ) > 0 and decreasing when f ′ ( x ) < 0 . Since x = − 3 , x = 0 and x = 3 are the partition numbers, the domain can be divided into four intervals as ( − ∞ , − 3 ) , ( − 3 , 0 ) ( 0 , 3 ) and ( 3 , ∞ ) . Let x = − 4 be a test number in the interval ( − ∞ , − 3 ) . Substitute x = − 4 in f ′ ( x ) = 4 x 3 − 36 x . f ′ ( − 4 ) = 4 ( − 4 ) 3 − 36 ( − 4 ) = − 256 + 144 = − 112 < 0 Since, f ′ ( x ) < 0 , the function f ( x ) is decreasing in the interval ( − ∞ , − 3 ) . Let x = − 2 be a test number in the interval ( − 3 , 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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Chapter 11.1, Problem 55E

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To find: The intervals where the function is increasing or decreasing and to sketch the graph of f(x)=x4−18x2 and to determine the horizontal tangents.

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Chapter 11.1, Problem 54E

Chapter 11.1, Problem 56E

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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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The intervals where the function is increasing or decreasing and to sketch the graph of f ( x ) = x 4 − 18 x 2 and to determine the horizontal tangents.

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To find: The intervals where the function is increasing or decreasing and to sketch the graph of f(x)=x4−18x2 and to determine the horizontal tangents.

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Chapter 11 Solutions