Explanation The function is f ( x ) = − x 3 + 6 x 2 + 1 and interval is [ − 1 , 6 ] . Find the critical number in the interval [ − 1 , 6 ] to compute the absolute minimum and absolute maximum. Differentiate the function f ( x ) = − x 3 + 6 x 2 + 1 as follows. f ′ ( x ) = d d x ( − x 3 + 6 x 2 + 1 ) = − ( 3 ) x 3 − 1 + 6 ( 2 ) x 2 − 1 + ( 0 ) = − 3 x 2 + 12 x That is, f ′ ( x ) = − 3 x 2 + 12 x . Set f ′ ( x ) = 0 and solve for x . − 3 x 2 + 12 x = 0 x ( − 3 x + 12 ) = 0 That is, x = 0 or − 3 x + 12 = 0 . − 3 x + 12 = 0 x = − 12 − 3 = 4 Hence, the critical numbers are x = 0 and x = 4 . Find the values of the function at the critical points and the end points of the given interval. Substitute x = − 1 in f ( x ) = − x 3 + 6 x 2 + 1 , f ( − 1 ) = − ( − 1 ) 3 + 6 ( − 1 ) 2 + 1 = 1 + 6 + 1 = 8 That is, f ( − 1 ) = 8

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 14, Problem 11RE

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To find: The absolute extrema of f(x)=−x3+6x2+1 on the domain [−1,6] if exist.

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Chapter 14, Problem 10RE

Chapter 14, Problem 12RE

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Chapter 14 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

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 - Prob. 51E Ch. 14.1 - Prob. 53E Ch. 14.1 - Salmon Spawning The number of salmon swimming...Ch. 14.1 - Prob. 55E Ch. 14.1 - Prob. 56E Ch. 14.1 - Prob. 57E Ch. 14.1 - Prob. 58E Ch. 14.1 - Prob. 59E Ch. 14.1 - Prob. 60E Ch. 14.1 - Prob. 61E Ch. 14.1 - Prob. 62E Ch. 14.2 - Prob. 1YT Ch. 14.2 - Prob. 2YT Ch. 14.2 - Prob. 3YT Ch. 14.2 - Prob. 4YT Ch. 14.2 - Prob. 1WE Ch. 14.2 - Prob. 2WE Ch. 14.2 - Prob. 1E Ch. 14.2 - Prob. 2E Ch. 14.2 - Prob. 3E Ch. 14.2 - Prob. 4E Ch. 14.2 - Prob. 5E Ch. 14.2 - Prob. 6E Ch. 14.2 - Prob. 7E Ch. 14.2 - Prob. 8E Ch. 14.2 - Prob. 9E Ch. 14.2 - Prob. 10E Ch. 14.2 - Prob. 11E Ch. 14.2 - Prob. 12E Ch. 14.2 - Prob. 13E Ch. 14.2 - Prob. 14E Ch. 14.2 - Revenue A local club is arranging a charter flight...Ch. 14.2 - Prob. 16E Ch. 14.2 - Prob. 17E Ch. 14.2 - Prob. 18E Ch. 14.2 - Prob. 19E Ch. 14.2 - Prob. 20E Ch. 14.2 - Prob. 21E Ch. 14.2 - Prob. 22E Ch. 14.2 - Prob. 23E Ch. 14.2 - Prob. 24E Ch. 14.2 - Prob. 25E Ch. 14.2 - Container Design Your company needs to design...Ch. 14.2 - 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Prob. 8E Ch. 14.3 - Prob. 9E Ch. 14.3 - Prob. 10E Ch. 14.3 - Prob. 11E Ch. 14.3 - Prob. 12E Ch. 14.3 - Prob. 13E Ch. 14.3 - Prob. 14E Ch. 14.3 - Prob. 15E Ch. 14.3 - Prob. 16E Ch. 14.3 - Prob. 17E Ch. 14.3 - Prob. 18E Ch. 14.3 - Elasticity For each of the following demand...Ch. 14.3 - Elasticity For each of the following demand...Ch. 14.3 - Prob. 21E Ch. 14.3 - Elasticity For each of the following demand...Ch. 14.3 - Prob. 23E Ch. 14.3 - Prob. 24E Ch. 14.3 - Prob. 25E Ch. 14.3 - Prob. 26E Ch. 14.3 - Prob. 27E Ch. 14.3 - Prob. 28E Ch. 14.3 - Elasticity of Software The Valve Corporation, a...Ch. 14.3 - Prob. 30E Ch. 14.3 - Elasticity A study of the demand for air travel in...Ch. 14.3 - Elasticity The price along the West Coast of the...Ch. 14.3 - Prob. 33E Ch. 14.3 - Prob. 34E Ch. 14.4 - Prob. 1YT Ch. 14.4 - Prob. 2YT Ch. 14.4 - The graph of y4 x4 y2 + x2 = 0 is called the...Ch. 14.4 - Prob. 4YT Ch. 14.4 - Prob. 1WE Ch. 14.4 - Prob. 2WE Ch. 14.4 - Find dy/dx by implicit differentiation for the...Ch. 14.4 - 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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 - Prob. 40RE Ch. 14 - Prob. 41RE Ch. 14 - Prob. 42RE Ch. 14 - Prob. 43RE Ch. 14 - Prob. 44RE Ch. 14 - Prob. 45RE Ch. 14 - Prob. 46RE Ch. 14 - Prob. 47RE Ch. 14 - Prob. 48RE Ch. 14 - Prob. 49RE Ch. 14 - Prob. 50RE Ch. 14 - Prob. 51RE Ch. 14 - Prob. 52RE Ch. 14 - Prob. 53RE Ch. 14 - Prob. 54RE Ch. 14 - Prob. 55RE Ch. 14 - Prob. 56RE Ch. 14 - Prob. 57RE Ch. 14 - Prob. 58RE Ch. 14 - Prob. 59RE Ch. 14 - Prob. 60RE Ch. 14 - Prob. 61RE Ch. 14 - Prob. 62RE Ch. 14 - Prob. 63RE Ch. 14 - Prob. 64RE Ch. 14 - Prob. 65RE Ch. 14 - Prob. 66RE Ch. 14 - Prob. 67RE Ch. 14 - Prob. 68RE

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The absolute extrema of f ( x ) = − x 3 + 6 x 2 + 1 on the domain [ − 1 , 6 ] if exist.

Solution Summary: The author explains how to find the absolute extrema of f(x) on the domain

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To find: The absolute extrema of f(x)=−x3+6x2+1 on the domain [−1,6] if exist.

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Chapter 14 Solutions