Explanation Given information: The function is f ( x , y ) = 6 x y 2 − 2 x 3 − 3 y 4 and the points are ( 0 , 0 ) , ( 1 , 1 ) , and ( 1 , − 1 ) . Explanation: Use the Second-Derivative test to determine the nature of the f ( x , y ) = 6 x y 2 − 2 x 3 − 3 y 4 at the points ( 0 , 0 ) , ( 1 , 1 ) , and ( 1 , − 1 ) . The Second-Derivative test: Suppose that f ( x , y ) is a function and ( a , b ) is a point at which ∂ f ∂ x ( a , b ) = 0 and ∂ f ∂ y ( a , b ) = 0 and D ( x , y ) = ∂ 2 f ∂ x 2 × ∂ 2 f ∂ y 2 − ( ∂ 2 f ∂ x ∂ y ) 2 . There are four conditions to determine a relative maximum and minimum as follows: 1. If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) > 0 , then f ( x , y ) has a relative minimum at ( a , b ) . 2. If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) < 0 , then f ( x , y ) has a relative maximum at ( a , b ) . 3. If D ( a , b ) < 0 , then f ( x , y ) has neither a relative maximum nor a relative minimum at ( a , b ) . 4. If D ( a , b ) = 0 , then no conclusion can be drawn from this test. Differentiate f ( x , y ) = 6 x y 2 − 2 x 3 − 3 y 4 partially twice with respect to x and y , it becomes ∂ f ∂ x = ∂ ∂ x ( 6 x y 2 − 2 x 3 − 3 y 4 ) ⇒ ∂ f ∂ x = 6 y 2 − 6 x 2 ∂ ∂ x ( ∂ f ∂ x ) = ∂ ∂ x ( 6 y 2 − 6 x 2 ) ⇒ ∂ 2 f ∂ x 2 = − 12 x ∂ f ∂ y = ∂ ∂ y ( 6 x y 2 − 2 x 3 − 3 y 4 ) ⇒ ∂ f ∂ y = 12 x y − 12 y 3 ∂ ∂ y ( ∂ f ∂ y ) = ∂ ∂ y ( 12 x y − 12 y 3 ) ⇒ ∂ 2 f ∂ y 2 = 12 x − 36 y 2 Now, differentiate ∂ f ∂ y = 12 x y − 12 y 3 with respect to x by treating y as constant. ∂ ∂ x ( ∂ f ∂ y ) = ∂ ∂ x ( 12 x y − 12 y 3 ) ⇒ ∂ 2 f ∂ x ∂ y = 12 y Substitute these values of ∂ 2 f ∂ x 2 = − 12 x , ∂ 2 f ∂ y 2 = 12 x − 36 y 2 , and ∂ 2 f ∂ x ∂ y = 12 y in the formula D ( x , y ) = ∂ 2 f ∂ x 2 × ∂ 2 f ∂ y 2 − ( ∂ 2 f ∂ x ∂ y ) 2

15th Edition

ISBN: 9780137590896

Author: Larry J. Goldstein; David C. Lay; David I. Schneider; Nakhle H. Asmar; William Edward Tavernetti

Publisher: Pearson Education (US)

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Question

Chapter 7.3, Problem 20E

To determine

The nature of f(x, y)=6xy2−2x3−3y4 at given points (0, 0),(1, 1), and (1, −1) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

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Chapter 7.3, Problem 19E

Chapter 7.3, Problem 21E

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Chapter 7 Solutions

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The nature of f ( x , y ) = 6 x y 2 − 2 x 3 − 3 y 4 at given points ( 0 , 0 ) , ( 1 , 1 ) , and ( 1 , − 1 ) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

Solution Summary: The author explains how the second-derivative test determines the nature of f(x,y)=6x

The nature of f(x, y)=6xy2−2x3−3y4 at given points (0, 0),(1, 1), and (1, −1) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

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Chapter 7 Solutions