Explanation Given: The function is f ( x ) = − x 2 + 4 x − 8 . The interval is [ − 4 , 4 ] . Calculation: According to the extreme-value theorem, it needs to consider only the critical numbers of f ( x ) and the endpoints. Endpoints are − 4 and 4 . Now find the derivative and determine the critical numbers: The given function is, f ( x ) = − x 2 + 4 x − 8 Differentiate the above function with respect to x . f ′ ( x ) = − 2 x + 4 To find the critical numbers, put g ′ ( x ) = 0 . This implies, − 2 x + 4 = 0 Simplify the above expression. − 2 x + 4 = 0 2 x = 4 x = 4 2 x = 2 Therefore, x = 2 . The only critical number in the interval [ − 4 , 4 ] is x = 2 . The absolute maximum or minimum must occur at endpoints or critical numbers − that is − 4 , 2 , 4 . Evaluate f ( x ) at these three numbers to see which ones give the largest and smallest values: Substitute 2 , − 4 , 4 for x in the equation of f ( x )

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Author: MULLINS

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Chapter 12.3, Problem 2CP

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To calculate: The absolute maximum and absolute minimum for given function on the interval [−4,4].

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Chapter 12.3, Problem 1CP

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Chapter 12 Solutions

EBK MATHEMATICS WITH APPLICATIONS IN TH

Ch. 12.1 - Checkpoint 1 For what values of x is the function...Ch. 12.1 - Checkpoint 2 Find all intervals on which is...Ch. 12.1 - Checkpoint 3 Identity the x-values of all points...Ch. 12.1 - Checkpoint 4 Find the critical numbers for each of...Ch. 12.1 - Prob. 5CP Ch. 12.1 - Prob. 6CP Ch. 12.1 - Checkpoint 7 Find the locations of the local...Ch. 12.1 - Prob. 8CP Ch. 12.1 - Checkpoint 9 If a sales function is given by...Ch. 12.1 - Prob. 1E

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The absolute maximum and absolute minimum for given function on the interval [ − 4 , 4 ] .

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