To determine To calculate: The location of local extremum of function, f ( x ) = − 3 x 4 − 4 x 3 + 12 x 2 − 1 . Explanation Given: The function is f ( x ) = − 3 x 4 − 4 x 3 + 12 x 2 − 1 . Formula Used: If f is function, then a number c for which f ( c ) is defined and e i t h e r f ' ( 0 ) = 0 . The function f is said to have a local extremum at c if it has a local maximum or minimum there. If f has local extremum at x = c , then c is a critical number of f . Calculation: Consider the provided function: f ( x ) = − 3 x 4 − 4 x 3 + 12 x 2 − 1 Differentiate the above function with respect to x , f ′ ( x ) = − 3 ( 4 x 3 ) − 4 ( 3 x 2 ) + 12 ( 2 x ) = − 12 x 3 − 12 x 2 + 24 x Solve for f ′ ( x ) = 0 to find the critical points, − 12 x 3 − 12 x 2 + 24 x = 0 Simplify and solve, − 12 x 3 − 12 x 2 + 24 x = 0 − 12 x ( x 2 + 1 x − 2 x ) = 0 − 12 x ( x 2 + ( 2 − 1 ) x − 2 x ) = 0 − 12 x ( x 2 + 2 x − 1 x − 2 x ) = 0 Simplify further, − 12 x ( x ( x + 2 ) − 1 ( x + 2 ) ) = 0 − 12 x ( x + 2 ) ( x − 1 ) = 0 Solution of above equation are as follows: x = 0 And, x + 2 = 0 x = − 2 And, x − 1 = 0 x = 1 Use the first derivative test: Take points as x = − 2

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 12.1, Problem 28E

To determine

To calculate: The location of local extremum of function, f(x)=−3x4−4x3+12x2−1.

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Chapter 12.1, Problem 27E

Chapter 12.1, Problem 29E

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Chapter 12 Solutions

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

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