Explanation Given information: The function is f ( x , y ) = 2 x y + y 2 + 2 x − 1 . Explanation: If f ( x , y ) has either a relative maximum or minimum at point ( x , y ) = ( a , b ) , then ∂ f ∂ x ( a , b ) = 0 and ∂ f ∂ y ( a , b ) = 0 . To find the partial derivatives ∂ f ∂ x and ∂ f ∂ y , differentiate ∂ f ∂ x = ∂ ∂ x ( 2 x y + y 2 + 2 x − 1 ) with respect to x . ⇒ ∂ f ∂ x = 2 y + 2 Now, differentiate the given function with respect to y . ∂ f ∂ y = ∂ ∂ y ( 2 x y + y 2 + 2 x − 1 ) ⇒ ∂ f ∂ y = 2 x + 2 y . Now, set ∂ f ∂ x = 0 and ∂ f ∂ y = 0 , and solve for x and y . Therefore, 2 y + 2 = 0 and 2 x + 2 y = 0 . As 2 y + 2 = 0 that implies y = − 1 . By substituting y = − 1 in 2 x + 2 y = 0 , ⇒ 2 x + 2 ( − 1 ) = 0 By simplifying, ⇒ 2 x − 2 = 0 ⇒ x = 1 Therefore, the function f ( x , y ) = 2 x y + y 2 + 2 x − 1 has a possible relative maximum or minimum at ( 1 , − 1 ) . Now, by the second derivative test there are four conditions to determine a relative maximum and minimum. A point ( a , b ) is relative minimum of f ( x , y ) , if D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) > 0 . A point ( a , b ) is relative maximum of f ( x , y ) , if D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) < 0 . There is no relative maximum or minimum of f ( x , y ) at point ( a , b ) , if D ( a , b ) < 0 . If D ( a , b ) = 0 , then no conclusion can be drawn from this test, where D ( x , y ) = ∂ 2 f ∂ x 2 × ∂ 2 f ∂ y 2 − ( ∂ 2 f ∂ x ∂ y ) 2

12th Edition

ISBN: 9780137590810

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

Publisher: PEARSON

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Chapter 7.3, Problem 32E

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All the points (x, y) where the function f(x, y)=2xy+y2+2x−1 has a possible relative maximum or minimum, and the nature of f(x, y) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

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Chapter 7.3, Problem 31E

Chapter 7.3, Problem 33E

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All the points ( x , y ) where the function f ( x , y ) = 2 x y + y 2 + 2 x − 1 has a possible relative maximum or minimum, and the nature of f ( x , y ) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

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All the points (x, y) where the function f(x, y)=2xy+y2+2x−1 has a possible relative maximum or minimum, and the nature of f(x, y) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

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