Explanation Given information: The function is f ( x , y ) = 2 x 2 − x 4 − y 2 and the points ( − 1 , 0 ) , ( 0 , 0 ) and ( 1 , 0 ) . Explanation: Use the Second-Derivative test to determine the nature of the f ( x , y ) = 2 x 2 − x 4 − y 2 at the points ( − 1 , 0 ) , ( 0 , 0 ) and ( 1 , 0 ) . 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 ) = 2 x 2 − x 4 − y 2 partially twice with respect to x and y , it becomes ∂ f ∂ x = ∂ ∂ x ( 2 x 2 − x 4 − y 2 ) ⇒ ∂ f ∂ x = 4 x − 4 x 3 ∂ ∂ x ( ∂ f ∂ x ) = ∂ ∂ x ( 4 x − 4 x 3 ) ⇒ ∂ 2 f ∂ x 2 = 4 − 12 x 2 ∂ f ∂ y = ∂ ∂ y ( 2 x 2 − x 4 − y 2 ) ⇒ ∂ f ∂ y = − 2 y ∂ ∂ y ( ∂ f ∂ y ) = ∂ ∂ y ( − 2 y ) ⇒ ∂ 2 f ∂ y 2 = − 2 Now, differentiate ∂ f ∂ y = − 2 y with respect to x by treating y as constant. ∂ ∂ x ( ∂ f ∂ y ) = ∂ ∂ x ( − 2 y ) ⇒ ∂ 2 f ∂ x ∂ y = 0 Substitute these values of ∂ 2 f ∂ x 2 = 4 − 12 x 2 , ∂ 2 f ∂ y 2 = − 2 , and ∂ 2 f ∂ x ∂ y = 0 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 21E

To determine

The nature of f(x, y)=2x2−x4−y2 at given points (−1, 0),(1, 0), and (1, 0) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

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Chapter 7.3, Problem 20E

Chapter 7.3, Problem 22E

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

Solution Summary: The author explains the second-derivative test to determine the nature of f(x,y)=2x2-4-

The nature of f(x, y)=2x2−x4−y2 at given points (−1, 0),(1, 0), and (1, 0) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

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Chapter 7 Solutions