Explanation Given Information: The provided function is g ( x ) = 1 x − 1 x 2 . Calculation: Consider the provided function, g ( x ) = 1 x − 1 x 2 Since, the domain of the function is not provided, so consider it to be the set of all x for which g is defined which is x ≠ 0 . Differentiate both sides of the above equation with respect to x , g ′ ( x ) = − 1 x 2 + 2 x 3 Locating stationary points: Stationary Points: Points in the interior of the domain where the derivative of the function is zero. To locate stationary points set g ′ ( x ) = 0 and solve for x . So, g ′ ( x ) = 0 − 1 x 2 + 2 x 3 = 0 Add 1 x 2 to both sides of the above equation, − 1 x 2 + 2 x 3 = 0 − 1 x 2 + 2 x 3 + 1 x 2 = 1 x 2 2 x 3 = 1 x 2 2 x 2 = x 3 Subtracting x 3 from both sides of the equation to get: 2 x 2 − x 3 = x 3 − x 3 2 x 2 − x 3 = 0 Take x 2 common from both sides of the above equation, then 2 x 2 − x 3 = 0 x 2 ( 2 − x ) = 0 Thus, x 2 = 0 x = 0 Or 2 − x = 0 x = 2 The derivative g ′ ( x ) = 0 for x = 0 and x = 2 . Therefore, the stationary point of the function g ( x ) = 1 x − 1 x 2 is x = 0 and x = 2 . Locating Singular points: Singular Points: Points in the interior of the domain where the derivative of the function is not defined. To locate singular points, find the interior points x where g ′ ( x ) is not defined but g ( x ) is defined. The function g ′ ( x ) = − 1 x 2 + 2 x 3 is not defined for x = 0 and x = 0 is not in the domain. Therefore, the function g ′ ( x ) = − 1 x 2 + 2 x 3 is defined for all x in the domain

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 12.1, Problem 34E

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To calculate: The exact location of the relative and absolute extrema of the function g(x)=1x−1x2.

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Chapter 12.1, Problem 33E

Chapter 12.1, Problem 35E

Chapter 12 Solutions

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The exact location of the relative and absolute extrema of the function g ( x ) = 1 x − 1 x 2 .

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To calculate: The exact location of the relative and absolute extrema of the function g(x)=1x−1x2.

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Chapter 12 Solutions