Explanation Given information: The function is f ( x , y ) = 3 x 2 + 8 x y − 3 y 2 + 2 x + 6 y . 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 . Therefore, find the partial derivatives ∂ f ∂ x and ∂ f ∂ y of the given function. Differentiate f ( x , y ) = 3 x 2 + 8 x y − 3 y 2 + 2 x + 6 y partially with respect to x and y , it becomes ∂ f ∂ x = ∂ ∂ x ( 3 x 2 + 8 x y − 3 y 2 + 2 x + 6 y ) ⇒ ∂ f ∂ x = 6 x + 8 y + 2 ∂ f ∂ y = ∂ ∂ y ( 3 x 2 + 8 x y − 3 y 2 + 2 x + 6 y ) ⇒ ∂ f ∂ y = 8 x − 6 y + 6 . Now, set ∂ f ∂ x = 0 and ∂ f ∂ y = 0 and solve for x and y . Therefore, 6 x + 8 y + 2 = 0 and 8 x − 6 y + 6 = 0 . By solving the equation 8 x − 6 y + 6 = 0 for y . ⇒ 6 y = 8 x + 6 ⇒ y = 4 3 x + 1 By substituting y = 4 3 x + 1 in 6 x + 8 y + 2 = 0 , ⇒ 6 x + 8 ( 4 3 x + 1 ) + 2 = 0 By distributing 8 into parenthesis, ⇒ 6 x + 32 3 x + 8 + 2 = 0 By combining like terms, ⇒ 50 3 x + 10 = 0 ⇒ 50 3 x = − 10 ⇒ x = − 3 5 Substitute x = − 3 5 in y = 4 3 x + 1 and simplify. ⇒ y = 4 3 ( − 3 5 ) + 1 ⇒ y = − 4 5 + 5 5 = 1 5 Therefore, the function f ( x , y ) = 3 x 2 + 8 x y − 3 y 2 + 2 x + 6 y has a possible relative maximum or minimum at ( − 3 5 , 1 5 ) . Now, by the second derivative test, there are four conditions to determine a relative maximum and minimum. If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) > 0 , then f ( x , y ) has a relative minimum at ( a , b ) . If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) < 0 , then f ( x , y ) has a relative maximum at ( a , b )

Calculus & Its Applications (14th Edition)

14th Edition

ISBN: 9780134437774

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

Publisher: PEARSON

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Chapter 7.3, Problem 27E

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All the points (x, y) where the function f(x, y)=3x2+8xy−3y2+2x+6y 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 26E

Chapter 7.3, Problem 28E

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Chapter 7 Solutions

Calculus & Its Applications (14th Edition)

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All the points ( x , y ) where the function f ( x , y ) = 3 x 2 + 8 x y − 3 y 2 + 2 x + 6 y 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 analyzes the function f(x,y)=3x2+8xy+2x+6y by using second-derivative test.

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All the points (x, y) where the function f(x, y)=3x2+8xy−3y2+2x+6y 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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