Explanation Given Information: The provided equation is g ( x ) = x 4 − 4 x . Formula used: First Derivative Test is, assume that for a critical point c of the continuous function f , its derivative is defined for x close to, and on both sides of x = c . Then, determine the sign of the derivative to the left and the right of x = c . If f ' ( x ) is positive and negative to left and right of x = c respectively, then at the point x = c , f has a maximum. If f ' ( x ) is negative and positive to left and right of x = c respectively, then at the point x = c , f has a minimum. If f ' ( x ) has either both positive or negative at x = c , then f has neither a maximum nor a minimum at the point x = c . Stationary Points are the points in the interior of the domain where the derivative is zero. Singular Points are the points in the interior of the domain where the derivative is not defined. Closed intervals contain end points which are the end points of the functions and the function does not have end points if the interval is open as open intervals do not have any end points. Calculation: Consider the provided equation: f ( x ) = x 4 − 4 x Differentiate both sides of the equation with respect to x . g ' ( x ) = 4 x 3 − 4 Now, locate stationary points. Recall that stationary Points are the points in the interior of the domain where the derivative is zero. Solve the equation g ' ( x ) = 0 for x . g ' ( x ) = 0 4 x 3 − 4 = 0 Factor 4 x 3 − 4 = 0 . 4 x 3 − 4 = 4 ( x 3 − 1 ) Thus, 4 ( x 3 − 1 ) = 0 . Consider 4 ( x 3 − 1 ) = 0 . Solve for x . 4 ( x 3 − 1 ) = 0 x 3 − 1 = 0 x 3 = 1 x = 1 The domain of the function is [ − 1 , 1 ] , so x = 1 is in the interior of 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, Problem 3RE

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To calculate: The exact location of the relative and absolute extrema of the function g(x)=x4−4x with domain [−1,1].

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Chapter 12, Problem 2RE

Chapter 12, Problem 4RE

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The exact location of the relative and absolute extrema of the function g ( x ) = x 4 − 4 x with domain [ − 1 , 1 ] .

Solution Summary: The author calculates the exact location of the relative and absolute extrema of a function g(x)=x 4-4x with domain.

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To calculate: The exact location of the relative and absolute extrema of the function g(x)=x4−4x with domain [−1,1].

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Chapter 12 Solutions