Explanation Given: Function is f ( x ) = 1 3 x 3 − 1 2 x 2 − 6 x + 3 and specified domain is [ − 4 , 4 ] . Approach: Steps for calculating absolute extrema in given interval are given below: First we find critical numbers of function in given interval. (Critical numbers are that values where f ' ( x ) = 0 ). Now we plugin each critical number from step 1 in given function. We also plugin end points of given interval into given function. The largest value which we get is absolute maximum and smallest value is absolute minimum. Calculation: Given function is f ( x ) = 1 3 x 3 − 1 2 x 2 − 6 x + 3 . On differentiating with respect to x f ' ( x ) = d d x ( 1 3 x 3 − 1 2 x 2 − 6 x + 3 ) = x 2 − x − 6 For critical values we can make derivative equal to zero. f ' ( x ) = 0 x 2 − x − 6 = 0 x ( x − 3 ) + 2 ( x − 3 ) = 0 ( x − 3 ) ( x + 2 ) = 0 x − 3 = 0 or x + 2 = 0 x = 3 or x = − 2 So critical points are x = − 2 , 3 . So both critical points are in given interval [ − 4 , 4 ] . Value of function at x = − 2 is f ( − 2 ) = 1 3 ( − 2 ) 3 − 1 2 ( − 2 ) 2 − 6 ( − 2 ) + 3 = − 8 3 + 13 = 31 3 = 10

2nd Edition

ISBN: 9780321964038

Author: GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher: Pearson Addison Wesley,

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Chapter 6.1, Problem 14E

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Value of all extrema for the given function f(x)=13x3−12x2−6x+3 in a specified domain [−4,4] if exists.

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Chapter 6.1, Problem 13E

Chapter 6.1, Problem 15E

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Value of all extrema for the given function f ( x ) = 1 3 x 3 − 1 2 x 2 − 6 x + 3 in a specified domain [ − 4 , 4 ] if exists.

Solution Summary: The author explains the steps for calculating absolute extrema in given interval.

Concept explainers

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To find:

Value of all extrema for the given function f(x)=13x3−12x2−6x+3 in a specified domain [−4,4] if exists.

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Chapter 6 Solutions

Value of all extrema for the given function f ( x ) = 1 3 x 3 − 1 2 x 2 − 6 x + 3 in a specified domain [ − 4 , 4 ] if exists.

Solution Summary: The author explains the steps for calculating absolute extrema in given interval.

Concept explainers

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To find: Value of all extrema for the given function f(x)=13x3−12x2−6x+3 in a specified domain [−4,4] if exists.

Want to see the full answer?

Chapter 6 Solutions

To find:

Value of all extrema for the given function f(x)=13x3−12x2−6x+3 in a specified domain [−4,4] if exists.