Explanation Given information: Given function is w ( t ) = t 4 − 1 t Calculation: A function f ( x ) is said to be concave up when f ' ' ( x ) > 0 and is said to be concave down when f ' ' ( x ) < 0 Applying derivative to the given function twice with respect to t : w ( t ) = t 4 − 1 t w ' ( t ) = ( t ) ( 4 t 3 − 0 ) − ( t 4 − 1 ) ( 1 ) t 2 w ' ( t ) = 4 t 4 − t 4 + 1 t 2 w ' ( t ) = 3 t 4 + 1 t 2 w ' ' ( t ) = ( t 2 ) ( 3 ( 4 t 3 ) + 0 ) − ( 3 t 4 + 1 ) ( 2 t ) ( t 2 ) 2 w ' ' ( t ) = ( t 2 ) ( 12 t 3 ) − ( 6 t 5 + 2 t ) t 4 w ' ' ( t ) = 12 t 5 − 6 t 5 − 2 t t 4 w ' ' ( t ) = 6 t 5 − 2 t t 4 w ' ' ( t ) = 6 t 4 − 2 t 3 The function is concave up when w ' ' ( t ) > 0 and concave down when w ' ' ( t ) < 0 The domain of the function can be written as the intervals: − ∞ < t < − 1 3 4 , − 1 3 4 < t < 0 , 0 < t < 1 3 4 , 1 3 4 < t < ∞ Since the function is not defined at t = 0 and the second derivative has zeroes at t = − 1 3 4 , 1 3 4 . Checking the value of the second derivative in each of the intervals: For − ∞ < t < − 1 3 4 , let t = − 1 w ' ' ( t ) = 6 t 4 − 2 t 3 w ' ' ( − 1 ) = 6 ( − 1 ) 4 − 2 ( − 1 ) 3 w ' ' ( − 1 ) = 6 ( 1 ) − 2 ( − 1 ) w ' ' ( − 1 ) = 6 − 2 − 1 w ' ' ( − 1 ) = − 4 Hence the function is concave down in the interval − ∞ < t < − 1 3 4 . For − 1 3 4 < t < 0 , let t = − 0.5 w ' ' ( t ) = 6 t 4 − 2 t 3 w ' ' ( − 0.5 ) = 6 ( − 0.5 ) 4 − 2 ( − 0.5 ) 3 w ' ' ( − 0.5 ) = 6 ( 0.0625 ) − 2 ( − 0.125 ) w ' ' ( − 0.5 ) = 0.375 − 2 − 0.125 w ' ' ( − 0.5 ) = − 1.625 − 0.125 w ' ' ( − 0.5 ) = 13 Hence the function is concave up in the interval − 1 3 4 < t < 0 . For 0 < t < 1 3 4 , let t = 0

4th Edition

ISBN: 9781319379421

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 4.4, Problem 18E

To determine

To find the intervals on which the function is concave up or down and the points of inflection.

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Chapter 4.4, Problem 17E

Chapter 4.4, Problem 19E

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Chapter 4 Solutions

CALCULUS 4E (HC) W/ ACHIEVE ACCESS

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Prob. 3E Ch. 4.8 - Prob. 4E Ch. 4.8 - Prob. 5E Ch. 4.8 - Prob. 6E Ch. 4.8 - Prob. 7E Ch. 4.8 - Prob. 8E Ch. 4.8 - Prob. 9E Ch. 4.8 - Prob. 10E Ch. 4.8 - Prob. 11E Ch. 4.8 - Prob. 12E Ch. 4.8 - Prob. 13E Ch. 4.8 - Prob. 14E Ch. 4.8 - Prob. 15E Ch. 4.8 - Prob. 16E Ch. 4.8 - Prob. 17E Ch. 4.8 - Prob. 18E Ch. 4.8 - Prob. 19E Ch. 4.8 - Prob. 20E Ch. 4.8 - Prob. 21E Ch. 4.8 - Prob. 22E Ch. 4.8 - Prob. 23E Ch. 4.8 - Prob. 24E Ch. 4.8 - Prob. 25E Ch. 4.8 - Prob. 26E Ch. 4.8 - Prob. 27E Ch. 4.8 - Prob. 28E Ch. 4.8 - Prob. 29E Ch. 4.8 - Prob. 30E Ch. 4.8 - Prob. 31E Ch. 4.8 - Prob. 32E Ch. 4.8 - Prob. 33E Ch. 4.8 - Prob. 34E Ch. 4.8 - Prob. 35E Ch. 4.8 - Prob. 36E Ch. 4.8 - Prob. 37E Ch. 4.8 - Prob. 38E Ch. 4.8 - Prob. 39E Ch. 4.8 - Prob. 40E Ch. 4.8 - Prob. 41E Ch. 4 - Prob. 1CRE Ch. 4 - Prob. 2CRE Ch. 4 - Prob. 3CRE Ch. 4 - Prob. 4CRE Ch. 4 - Prob. 5CRE Ch. 4 - Prob. 6CRE Ch. 4 - Prob. 7CRE Ch. 4 - Prob. 8CRE Ch. 4 - Prob. 9CRE Ch. 4 - Prob. 10CRE Ch. 4 - Prob. 11CRE Ch. 4 - Prob. 12CRE Ch. 4 - 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Prob. 63CRE Ch. 4 - Prob. 64CRE Ch. 4 - Prob. 65CRE Ch. 4 - Prob. 66CRE Ch. 4 - Prob. 67CRE Ch. 4 - Prob. 68CRE Ch. 4 - Prob. 69CRE Ch. 4 - Prob. 70CRE Ch. 4 - Prob. 71CRE Ch. 4 - Prob. 72CRE Ch. 4 - Prob. 73CRE Ch. 4 - Prob. 74CRE Ch. 4 - Prob. 75CRE Ch. 4 - Prob. 76CRE Ch. 4 - Prob. 77CRE Ch. 4 - Prob. 78CRE Ch. 4 - Prob. 79CRE Ch. 4 - Prob. 80CRE Ch. 4 - Prob. 81CRE Ch. 4 - Prob. 82CRE Ch. 4 - Prob. 83CRE Ch. 4 - Prob. 84CRE Ch. 4 - Prob. 85CRE Ch. 4 - Prob. 86CRE Ch. 4 - Prob. 87CRE Ch. 4 - Prob. 88CRE Ch. 4 - Prob. 89CRE

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