Explanation Given information: The function given is, f ( x ) = x ln x Formula used: The derivative of the exponential function is the exponential function. d ( e x ) d x = e x Calculation: The first derivative of the function is determined first. d d x ( f ( x ) ) = d d x ( x ln x ) = ln x × 1 − x × 1 x ( ln x ) 2 = ln x − 1 ( ln x ) 2 f ' ( x ) = 1 ln x − 1 ( ln x ) 2 By differentiating the first derivative, the second derivative can be obtained as follows. d d x ( f ' ( x ) ) = d d x ( 1 ln x − 1 ( ln x ) 2 ) = ( − 1 ( ln x ) 2 × 1 x ) − ( − 2 ( ln x ) 3 × 1 x ) = 2 x ln 3 x − 1 x ln 2 x f ' ' ( x ) = 1 x ln 2 x ( 2 ln x − 1 ) As the definition of the exponential function, e x > 0 for any x

14th Edition

ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Chapter 3.3, Problem 48S

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To graph:The function given using the sign chart for f'' determining whether the function is concave up or concave down and identifying the inflection points.

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Chapter 3.3, Problem 47S

Chapter 3.3, Problem 49S

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find the zeros of the function algebraically: f(x) = 9x2 - 3x - 2

Chapter 3 Solutions

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To graph: The function given using the sign chart for f ' ' determining whether the function is concave up or concave down and identifying the inflection points.

Solution Summary: The author explains the function given using the sign chart. The first derivative of the exponential function is determined first.

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To graph:The function given using the sign chart for f'' determining whether the function is concave up or concave down and identifying the inflection points.

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Chapter 3 Solutions