Explanation Given: The function, f ( x ) = ( x − 2 ) 4 . Concept Used: A function is concave up on the interval where the second derivative is positive and concave down on the interval where second derivative is negative. Calculation: Consider the function, f ( x ) = ( x − 2 ) 4 In order to find the intervals of concavity, first find the inflection points by setting the second derivative of the function equal to zero. So, using the chain rule and power rule of derivatives, the first derivative of the function is given by f ' ( x ) = d d x ( x − 2 ) 4 = 4 ( x − 2 ) 4 − 1 ⋅ d d x ( x − 2 ) = 4 ( x − 2 ) 3 ⋅ ( 1 − 0 ) = 4 ( x − 2 ) 3 Again using the chain rule and power rule, the second derivative of the function is given by f ' ' ( x ) = d d x ( 4 ( x − 2 ) 3 ) = 4 ⋅ 3 ( x − 2 ) 3 − 1 ⋅ d d x ( x − 2 ) = 12 ( x − 2 ) 2 ⋅ ( 1 − 0 )

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 3.3, Problem 41S

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The intervals on which the function f(x)=(x−2)4 is concave up or concave down using a sign chart for f'' .

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Chapter 3.3, Problem 40S

Chapter 3.3, Problem 42S

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The intervals on which the function f ( x ) = ( x − 2 ) 4 is concave up or concave down using a sign chart for f ' ' .

Solution Summary: The author explains how a function is concave up and down on the intervals where the second derivative is positive and negative.

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The intervals on which the function f(x)=(x−2)4 is concave up or concave down using a sign chart for f'' .

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