Explanation Given Information: The provided function is: f ( x ) = x 3 − 12 x The interval is [ − 2 , + ∞ ) . Formula used: If f is a real-valued function, then its extrema occur among the following types of points: Stationary points: points which lie in the interior of the domain of a function. The derivative is zero at the stationary points. To locate the stationary points, equate f ' ( x ) to zero and solve for x . Singular points: points that lie in the interior of the domain of a function where the derivative is not defined. The interior points can be found by finding the values of x where f ' ( x ) is not defined. Endpoints: The end points of a function are the ending points of the domain. For a domain ( − a , a ) , the end points are − a and a . To locate possible points of inflection, list points where f ″ ( x ) = 0 and also interior points where f ″ ( x ) is not defined. Calculation: Consider the provided function: f ( x ) = x 3 − 12 x Now, calculate f ′ ( x ) as: f ′ ( x ) = 3 x 2 − 12 Now, to find stationary points, set the derivative equal to 0 and solve for x : 3 x 2 − 12 = 0 3 ( x 2 − 4 ) = 0 x = ± 2 Thus, there are two stationary points at x = − 2 and x = 2 . Use a test point to the right and left to determine that this stationary point is a relative maximum or minimum. For x = − 3 , f ( − 3 ) = ( − 3 ) 3 − 12 ( − 3 ) = − 27 + 36 = 9 For x = − 2 , f ( − 2 ) = ( − 2 ) 3 − 12 ( − 2 ) = − 8 + 24 = 16 For x = − 1 , f ( − 1 ) = ( − 1 ) 3 − 12 ( − 1 ) = − 1 + 12 = 11 For x = 2 , f ( 2 ) = ( 2 ) 3 − 12 ( 2 ) = 8 − 24 = − 16 Now, as there are no singular points of f ′ as f ( x ) is defined for all values of x

7th Edition

ISBN: 9781337291248

Author: Waner, Stefan.

Publisher: Cengage Learning,

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Chapter 5, Problem 17RE

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To calculate: The coordinates of the points of relative and absolute extrema, and the points of inflection, for the provided function f(x)=x3−12x on the interval, [−2,+∞).

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Chapter 5, Problem 16RE

Chapter 5, Problem 18RE

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The coordinates of the points of relative and absolute extrema, and the points of inflection, for the provided function f ( x ) = x 3 − 12 x on the interval, [ − 2 , + ∞ ) .

Solution Summary: The author calculates the coordinates of the relative and absolute extrema, and the points of inflection, for the provided function.

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To calculate: The coordinates of the points of relative and absolute extrema, and the points of inflection, for the provided function f(x)=x3−12x on the interval, [−2,+∞).

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