Explanation Given: The function is f ( x ) = − 4 3 x 3 + x 2 + 30 x − 7 . Calculation: Solve the given function: f ( x ) = − 4 3 x 3 + x 2 + 30 x − 7 Differentiate the above function: f ′ ( x ) = − 4 x 2 + 2 x + 30 Substitute 0 for f ′ ( x ) : − 4 x 2 + 2 x + 30 = 0 − 4 x 2 + 12 x − 10 x + 30 = 0 − 4 x ( x − 3 ) − 10 ( x − 3 ) = 0 ( − 4 x − 10 ) ( x − 3 ) = 0 The first critical number is: − 4 x − 10 = 0 x = − 5 2 The second critical number is: x − 3 = 0 x = 3 Thus, − 5 2 and 3 are the critical numbers for the function. Let choose a point − 3 less than − 5 2 and 0 greater than − 5 2 . Calculate the value of f ′ ( − 3 ) : f ′ ( − 3 ) = ( − 4 ( − 3 ) − 10 ) ( − 3 − 3 ) = ( 12 − 10 ) ( − 6 ) = 2 ( − 6 ) = − 12 Calculate the value of f ′ ( 0 ) : f ′ ( 0 ) = ( − 4 ( 0 ) − 10 ) ( 0 − 3 ) = ( − 10 ) ( − 3 ) = 30 Since, f ′ ( − 3 ) < 0 and f ′ ( 0 ) > 0 , − 5 2 is a point of local minima. Let choose a point 0 less than 3 and 4 greater than 3 . Calculate the value of f ′ ( 4 ) : f ′ ( 4 ) = ( − 4 ( 4 ) − 10 ) ( 4 − 3 ) = ( − 16 − 10 ) ( 1 ) = − 26 Since, f ′ ( 0 ) > 0 and f ′ ( 4 ) < 0 , 3 is a point of local maxima

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

12th Edition

ISBN: 9780134767628

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 12, Problem 22RE

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To calculate: The graph of the functions and list the location of local extremum and point of inflection, the intervals where the function is increasing and decreasing and the interval where it is concave upward and concave downward.

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Chapter 12, Problem 21RE

Chapter 12, Problem 23RE

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Chapter 12 Solutions

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

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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