Explanation Given information: The provided function is: g ( x ) = x 3 − 12 x Formula used: Stationary Points: To locate stationary points put f ′ ( x ) = 0 and solve for x . Second derivative test: If the function f ( x ) has a stationary point at x = c and that f ″ ( c ) exists then evaluate the sign of f ″ ( c ) : A. If f ″ ( c ) < 0 , then f has a relative maximum at x = c . B. If f ″ ( c ) > 0 , then f has a relative minimum at x = c . Also, if f ″ ( c ) = 0 then the test is fails so use the first derivative test. Calculation: Consider the provided function, g ( x ) = x 3 − 12 x . Differentiate the above equation with respect to x to get, g ′ ( x ) = d d x ( x 3 − 12 x ) = 3 x 2 − 12 Thus, g ′ ( x ) = 3 x 2 − 12 Substitute 3 x 2 − 12 for g ′ ( x ) in g ′ ( x ) = 0 to get the stationary points, 3 x 2 − 12 = 0 Solve the above equation for x to get: 3 x 2 = 12 x 2 = 12 3 x 2 = 4 x = ± 2 Thus, x = 2 and x = − 2 Since, g ′ ( x ) = 0 at x = 2 , − 2

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 12.3, Problem 35E

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To calculate: The x-coordinates of all the critical points of the function g(x)=x3−12x and check by first using the second derivative test and by some other method if the test fails, whether the each critical point is relative minimum, relative maximum or neither.

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Chapter 12.3, Problem 34E

Chapter 12.3, Problem 36E

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The x -coordinates of all the critical points of the function g ( x ) = x 3 − 12 x and check by first using the second derivative test and by some other method if the test fails, whether the each critical point is relative minimum, relative maximum or neither.

Solution Summary: The author calculates the x-coordinates of all the critical points of the function g(x)=x 3-12x and checks whether each critical point is relative minimum,

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To calculate: The x-coordinates of all the critical points of the function g(x)=x3−12x and check by first using the second derivative test and by some other method if the test fails, whether the each critical point is relative minimum, relative maximum or neither.

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