Explanation Given: function f ( x ) = x 2 ( x − 1 ) ( x − 2 ) 2 . Calculation: Find critical values by finding f ′ ( x ) as: f ( x ) = x 2 ( x − 1 ) ( x − 2 ) 2 f ′ ( x ) = x ( x 2 − 6 x + 4 ) ( x − 2 ) 3 Find the roots of f ′ ( x ) : x ( x 2 − 6 x + 4 ) ( x − 2 ) 3 = 0 x ( x 2 − 6 x + 4 ) = 0 x = 0 , x 2 − 6 x + 4 = 0 use formula to fnd roots x = − b ± b 2 − 4 a c 2 a x = − 6 + 6 2 − 4 ( 1 ) ( 4 ) 2 ( 1 ) = 3 + 5 x = − 6 − 6 2 − 4 ( 1 ) ( 4 ) 2 ( 1 ) = 3 − 5 The critical values are 0, 3 + 5 and 3 − 5 , which we will use to divide the a-axis into three intervals on the sign chart, the intervals are ( − ∞ , 0 ) , ( 0 , 3 − 5 ) , ( 3 − 5 , 3 + 5 ) and ( 3 + 5 , ∞ ) . Next find the sign of the derivative in each interval taking the value in each interval as: f ′ ( x ) = x ( x 2 − 6 x + 4 ) ( x − 2 ) 3 x = − 1 , f ′ ( 1 ) = − 1 ( ( − 1 ) 2 − 6 ( − 1 ) + 4 ) ( − 1 − 2 ) 3 = 0

14th Edition

ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Chapter 3.2, Problem 73S

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To sketch: Sketch the graph of a given function f .

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Chapter 3.2, Problem 72S

Chapter 3.2, Problem 74S

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Chapter 3 Solutions

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Sketch the graph of a given function f .

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Chapter 3 Solutions