Explanation Given information: The function is f ( x , y ) = x 4 − 8 x y + 2 y 2 − 3 . Explanation: Use First-Derivative test for functions of two variables: If f ( x , y ) has either a relative maximum or minimum at the point ( x , y ) = ( a , b ) , then ∂ f ∂ x ( a , b ) = 0 and ∂ f ∂ y ( a , b ) = 0 . Differentiate f ( x , y ) = x 4 − 8 x y + 2 y 2 − 3 partially with respect to x and y , it becomes ∂ f ∂ x = ∂ ∂ x ( x 4 − 8 x y + 2 y 2 − 3 ) ⇒ ∂ f ∂ x = 4 x 3 − 8 y ∂ f ∂ y = ∂ ∂ y ( x 4 − 8 x y + 2 y 2 − 3 ) ⇒ ∂ f ∂ y = − 8 x + 4 y By using the First-Derivative test, set first partial derivatives equal to zero and solve the system of equations. ∂ f ∂ x = 0 4 x 3 − 8 y = 0 ……………….. (1) Similarly, ∂ f ∂ y = 0 . − 8 x + 4 y = 0 ………………… (2) Solve the system of equations by using the substitution method as follows: Solve equation (2) for y

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Chapter 7.3, Problem 14E

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All possible relative maximum or minimum points (x, y) of the f(x, y)=x4−8xy+2y2−3.

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Chapter 7.3, Problem 13E

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All possible relative maximum or minimum points ( x , y ) of the f ( x , y ) = x 4 − 8 x y + 2 y 2 − 3 .

Solution Summary: The author explains the First-Derivative test for functions of two variables.

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All possible relative maximum or minimum points (x, y) of the f(x, y)=x4−8xy+2y2−3.

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