Explanation Given information: The function is f ( x , y ) = 2 x 2 + 3 x y + 5 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 . Therefore, find the partial derivatives ∂ f ∂ x and ∂ f ∂ y of the given function. By differentiating the given function with respect to x . ∂ f ∂ x = ∂ ∂ x ( 2 x 2 + 3 x y + 5 y 2 ) ⇒ ∂ f ∂ x = 4 x + 3 y By differentiating the given function with respect to y . ∂ f ∂ y = ∂ ∂ y ( 2 x 2 + 3 x y + 5 y 2 ) ⇒ ∂ f ∂ y = 3 x + 10 y . Now, set ∂ f ∂ x = 0 and ∂ f ∂ y = 0 and solve for x and y . Therefore, 4 x + 3 y = 0 and 3 x + 10 y = 0 . As 4 x + 3 y = 0 that implies x = − 3 4 y . By substituting x = − 3 4 y in 3 x + 10 y = 0 , ⇒ 3 ( − 3 4 y ) + 10 y = 0 By combining the like terms, ⇒ − 9 4 y + 10 y = 0 ⇒ ( − 9 4 + 40 4 ) y = 0 ⇒ 31 4 y = 0 ⇒ y = 0 As x = − 3 4 y , that implies x = 0 . Therefore, the function f ( x , y ) = 2 x 2 + 3 x y + 5 y 2 has a possible relative maximum or minimum at ( 0 , 0 ) . 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

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Chapter 7.3, Problem 38E

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All the points (x, y) where the function f(x, y)=2x2+3xy+5y2 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 37E

Chapter 7.3, Problem 39E

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All the points ( x , y ) where the function f ( x , y ) = 2 x 2 + 3 x y + 5 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) = 2x2+3xy+5. If the second-derivative test is inconclusive, then

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All the points (x, y) where the function f(x, y)=2x2+3xy+5y2 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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