Explanation Given information: The given function is, f ( x ) = 2 x x 2 + 1 Calculation: Use second derivative test to find the relative extrema. Compute both first and second derivatives, f ′ ( x ) and f ″ ( x ) . The first derivative of f ( x ) is, f ′ ( x ) = d d x ( 2 x x 2 + 1 ) = ( x 2 + 1 ) d d x ( 2 x ) − 2 x d d x ( x 2 + 1 ) ( x 2 + 1 ) 2 ( Use quotient rule of differentiation ) = ( x 2 + 1 ) 2 ( 1 x 1 − 1 ) − 2 x ( 2 x ) ( x 2 + 1 ) 2 = ( x 2 + 1 ) 2 ( 1 ) − 4 x 2 ( x 2 + 1 ) 2 Simplify it further. f ′ ( x ) = 2 ( x 2 + 1 ) − 4 x 2 ( x 2 + 1 ) 2 = 2 x 2 + 2 − 4 x 2 ( x 2 + 1 ) 2 = 2 ( 1 − x 2 ) ( x 2 + 1 ) 2 The second derivative of the function is, f ″ ( x ) = d d x f ′ ( x ) = d d x ( 2 ( 1 − x 2 ) ( x 2 + 1 ) 2 ) = ( x 2 + 1 ) 2 d d x 2 ( 1 − x 2 ) − 2 ( 1 − x 2 ) d d x ( x 2 + 1 ) 2 [ ( x 2 + 1 ) 2 ] 2 ( Use quotient rule ) = ( x 2 + 1 ) 2 2 ( 0 − 2 ( x 2 − 1 ) ) − 2 ( 1 − x 2 ) 2 ( x 2 + 1 ) 2 − 1 d d x ( x 2 + 1 ) ( x 2 + 1 ) 4 ( Use chain rule ) Simplify it further. f ″ ( x ) = ( x 2 + 1 ) 2 ( − 4 ) ( x ) − 2 ( 1 − x 2 ) 2 ( x 2 + 1 ) ( 2 x ) ( x 2 + 1 ) 4 = − 4 x ( x 2 + 1 ) 2 − 8 x ( 1 − x 2 ) ( x 2 + 1 ) ( x 2 + 1 ) 4 = − 4 x ( x 2 + 1 ) − 8 x ( 1 − x 2 ) ( x 2 + 1 ) 3 [ Cancel out common factor ( x 2 + 1 ) ] = − 4 x 3 − 4 x − 8 x + 8 x 3 ( x 2 + 1 ) 3 Combine the like terms

Applied Calculus for the Managerial, Life, and Social Sciences

10th Edition

ISBN: 9781305887831

Author: Tan

Publisher: CENGAGE CO

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Chapter 4.2, Problem 70E

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The relative extrema of the given function.

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Chapter 4.2, Problem 69E

Chapter 4.2, Problem 71E

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Chapter 4 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences

Ch. 4.1 - Explain each of the following: a. f is increasing...Ch. 4.1 - Prob. 2CQ Ch. 4.1 - Prob. 3CQ Ch. 4.1 - Prob. 4CQ Ch. 4.1 - Prob. 5CQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E

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The relative extrema of the given function.

Solution Summary: The author explains the relative extrema of the given function. The relative maximum and relative minimum are f(1)=1.

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The relative extrema of the given function.

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Chapter 4 Solutions