Explanation Given information: The function is f ( x , y ) = − 5 x 2 + 4 x y − 17 y 2 − 6 x + 6 y + 2 . 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 . Differentiating f ( x , y ) = − 5 x 2 + 4 x y − 17 y 2 − 6 x + 6 y + 2 partially with respect to x and y , ∂ f ∂ x = ∂ ∂ x ( − 5 x 2 + 4 x y − 17 y 2 − 6 x + 6 y + 2 ) ⇒ ∂ f ∂ x = − 10 x + 4 y − 6 ∂ f ∂ y = ∂ ∂ y ( − 5 x 2 + 4 x y − 17 y 2 − 6 x + 6 y + 2 ) ⇒ ∂ f ∂ y = 4 x − 34 y + 6 . Now, set ∂ f ∂ x = 0 and ∂ f ∂ y = 0 , and solve for x and y . Therefore, − 10 x + 4 y − 6 = 0 and 4 x − 34 y + 6 = 0 . By solving the equation − 10 x + 4 y − 6 = 0 for y . ⇒ 4 y = 10 x + 6 ⇒ y = 5 2 x + 3 2 By substituting y = 5 2 x + 3 2 in 4 x − 34 y + 6 = 0 , ⇒ 4 x − 34 ( 5 2 x + 3 2 ) + 6 = 0 By distributing − 34 into parenthesis, ⇒ 4 x − 85 x − 51 + 6 = 0 By combining the like terms, ⇒ − 81 x − 45 = 0 ⇒ 81 x = − 45 By dividing both the sides by 81 , ⇒ x = − 45 81 ⇒ x = − 5 9 Substitute x = − 5 9 in y = 5 2 x + 3 2 and simplify. ⇒ y = 5 2 ( − 5 9 ) + 3 2 ⇒ y = − 25 18 + 3 2 = − 25 + 27 18 ⇒ y = 2 18 = 1 9 Therefore, the function f ( x , y ) = − 5 x 2 + 4 x y − 17 y 2 − 6 x + 6 y + 2 has a possible relative maximum or minimum at ( − 5 9 , 1 9 ) . Now, by the second derivative test there are four conditions to determine a relative maximum and minimum. f ( x , y ) has a relative minimum at ( a , b ) , if D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) > 0 . f ( x , y ) has a relative maximum at ( a , b ) , if D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) < 0 . f ( x , y ) has neither a relative maximum nor a relative minimum at ( a , b ) , if D ( a , b ) < 0

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 25E

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All the points (x, y) where the function f(x, y)=−5x2+4xy−17y2−6x+6y+2 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 24E

Chapter 7.3, Problem 26E

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All the points ( x , y ) where the function f ( x , y ) = − 5 x 2 + 4 x y − 17 y 2 − 6 x + 6 y + 2 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.

Solution Summary: The author explains that the function f(x,y) has a possible relative maximum or minimum at the point . 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)=−5x2+4xy−17y2−6x+6y+2 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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