Explanation The function is f ( x ) = ( x 2 − 16 ) 2 3 and interval is [ − 5 , 8 ] . Find the critical number in the interval [ − 5 , 8 ] to compute the absolute minimum and absolute maximum. Differentiate the function f ( x ) = ( x 2 − 16 ) 2 3 as follows, f ′ ( x ) = d d x ( x 2 − 16 ) 2 3 = ( 2 3 ) ( x 2 − 16 ) 2 3 − 1 ( 2 x ) = ( 2 3 ) ( x 2 − 16 ) − 1 3 ( 2 x ) = 4 x 3 ( x 2 − 16 ) 1 3 Set f ′ ( x ) = 0 and solve for x. 4 x 3 ( x 2 − 16 ) 1 3 4 x = 0 x = 0 The derivative of the function is undefined when 3 ( x 2 − 16 ) 1 3 = 0 . That is, 3 ( x 2 − 16 ) 1 3 = 0 x = ± 4 The derivative is undefined when x = − 4 and x = 4 . Hence, the critical numbers are x = 0 , x = − 4 and x = 4 . Find the values of the function at the critical points and the end points of the given interval. Substitute x = − 5 in f ( x ) = ( x 2 − 16 ) 2 3 , f ( − 5 ) = ( ( − 5 ) 2 − 16 ) 2 3 = ( 25 − 16 ) 2 3 = 9 2 3 = 4 .33 That is, f ( − 5 ) = 4.33

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 14.1, Problem 22E

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To find: The absolute extrema of f(x)=(x2−16)23 on the domain [−5,8] if exist and verify the answer using the graphing calculator.

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Chapter 14.1, Problem 21E

Chapter 14.1, Problem 23E

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Chapter 14 Solutions

Finite Mathematics and Calculus with Applications

Ch. 14.1 - Find the absolute extrema of the function f(x) =...Ch. 14.1 - Find the locations and values of the absolute...Ch. 14.1 - Prob. 1WE Ch. 14.1 - Find all critical numbers for each of the...Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Find the locations of any absolute extrema for the...Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E

Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E Ch. 14.1 - Prob. 9E Ch. 14.1 - Prob. 10E Ch. 14.1 - Prob. 11E Ch. 14.1 - Prob. 12E Ch. 14.1 - Prob. 13E Ch. 14.1 - Prob. 14E Ch. 14.1 - Prob. 15E Ch. 14.1 - Prob. 16E Ch. 14.1 - Find the absolute extrema if they exist, as well...Ch. 14.1 - Prob. 18E Ch. 14.1 - Prob. 19E Ch. 14.1 - Prob. 20E Ch. 14.1 - Prob. 21E Ch. 14.1 - Prob. 22E Ch. 14.1 - Prob. 23E Ch. 14.1 - Prob. 24E Ch. 14.1 - Prob. 25E Ch. 14.1 - Prob. 26E Ch. 14.1 - Prob. 27E Ch. 14.1 - Prob. 28E Ch. 14.1 - Prob. 29E Ch. 14.1 - Prob. 30E Ch. 14.1 - Prob. 31E Ch. 14.1 - Prob. 32E Ch. 14.1 - Prob. 33E Ch. 14.1 - Prob. 34E Ch. 14.1 - Prob. 35E Ch. 14.1 - Find the absolute extrema if they exist, as well...Ch. 14.1 - Prob. 37E Ch. 14.1 - Prob. 38E Ch. 14.1 - Prob. 39E Ch. 14.1 - Prob. 40E Ch. 14.1 - Prob. 41E Ch. 14.1 - Prob. 42E Ch. 14.1 - Prob. 43E Ch. 14.1 - Prob. 44E Ch. 14.1 - Prob. 45E Ch. 14.1 - Prob. 46E Ch. 14.1 - Prob. 47E Ch. 14.1 - Prob. 48E Ch. 14.1 - Prob. 49E Ch. 14.1 - Prob. 50E Ch. 14.1 - 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Prob. 2E Ch. 14.6 - Prob. 3E Ch. 14.6 - Prob. 4E Ch. 14.6 - Prob. 5E Ch. 14.6 - For Exercises 18, find dy for the given values of...Ch. 14.6 - For Exercises 18, find dy for the given values of...Ch. 14.6 - Prob. 8E Ch. 14.6 - Prob. 9E Ch. 14.6 - Prob. 10E Ch. 14.6 - Prob. 11E Ch. 14.6 - Prob. 12E Ch. 14.6 - Prob. 13E Ch. 14.6 - Prob. 14E Ch. 14.6 - Use the differential to approximate each quantity....Ch. 14.6 - Prob. 16E Ch. 14.6 - Business and Economies 17.Demand The demand for...Ch. 14.6 - Prob. 18E Ch. 14.6 - Revenue A company estimates that the revenue (in...Ch. 14.6 - Prob. 20E Ch. 14.6 - Prob. 21E Ch. 14.6 - Prob. 22E Ch. 14.6 - Prob. 23E Ch. 14.6 - Prob. 24E Ch. 14.6 - Prob. 25E Ch. 14.6 - Prob. 26E Ch. 14.6 - Prob. 27E Ch. 14.6 - Prob. 28E Ch. 14.6 - Prob. 29E Ch. 14.6 - Gray Wolves Accurate methods of estimating the age...Ch. 14.6 - Prob. 31E Ch. 14.6 - Prob. 32E Ch. 14.6 - Prob. 33E Ch. 14.6 - Prob. 34E Ch. 14.6 - Prob. 35E Ch. 14.6 - Prob. 36E Ch. 14.6 - Prob. 37E Ch. 14.6 - Prob. 38E Ch. 14.6 - Prob. 39E Ch. 14.6 - Prob. 40E Ch. 14.6 - Prob. 41E Ch. 14 - The absolute maximum of a function always occurs...Ch. 14 - A continuous function on a closed interval has an...Ch. 14 - Prob. 3RE Ch. 14 - Prob. 4RE Ch. 14 - Total revenue is maximized at the price where...Ch. 14 - Prob. 6RE Ch. 14 - Prob. 7RE Ch. 14 - Prob. 8RE Ch. 14 - Prob. 9RE Ch. 14 - Prob. 10RE Ch. 14 - Prob. 11RE Ch. 14 - Prob. 12RE Ch. 14 - Prob. 13RE Ch. 14 - Prob. 14RE Ch. 14 - Prob. 15RE Ch. 14 - Prob. 16RE Ch. 14 - Prob. 17RE Ch. 14 - Prob. 18RE Ch. 14 - Prob. 19RE Ch. 14 - Prob. 20RE Ch. 14 - Prob. 21RE Ch. 14 - Prob. 22RE Ch. 14 - Prob. 23RE Ch. 14 - Prob. 24RE Ch. 14 - Prob. 25RE Ch. 14 - Prob. 26RE Ch. 14 - Prob. 27RE Ch. 14 - Prob. 28RE Ch. 14 - Prob. 29RE Ch. 14 - Prob. 30RE Ch. 14 - Prob. 31RE Ch. 14 - Why is implicit differentiation used in related...Ch. 14 - Prob. 33RE Ch. 14 - Prob. 34RE Ch. 14 - Prob. 35RE Ch. 14 - Prob. 36RE Ch. 14 - Prob. 37RE Ch. 14 - Prob. 38RE Ch. 14 - Prob. 39RE Ch. 14 - 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The absolute extrema of f ( x ) = ( x 2 − 16 ) 2 3 on the domain [ − 5 , 8 ] if exist and verify the answer using the graphing calculator.

Solution Summary: The author explains how to find the absolute extrema of f(x) and the critical number in the interval.

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To find: The absolute extrema of f(x)=(x2−16)23 on the domain [−5,8] if exist and verify the answer using the graphing calculator.

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Chapter 14 Solutions