(A) Explanation Result used: Let the partial derivatives f x x , f y y and f x y of the function z = f ( x , y ) exists in the circular region contained in the xy -plane with center ( a , b ) and f x ( a , b ) = 0 . (a) The function has local maximum at ( a , b ) if f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 > 0 and f x x ( a , b ) < 0 . (b) The function has local minimum at ( a , b ) if f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 > 0 and f x x ( a , b ) > 0 . (c) If f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 < 0 , then f ( a , b ) is a saddle point (neither a maximum nor a minimum). (d) If f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 = 0 , then the test fails. Proof: Consider the given function, f ( x , y ) = x 4 e y + x 2 y 4 + 1 . Calculate the partial derivative of the above function with respect to x . f x ( x , y ) = ∂ ∂ x ( x 4 e y + x 2 y 4 + 1 ) = ∂ ∂ x ( x 4 e y ) + ∂ ∂ x ( x 2 y 4 ) + ∂ ∂ x ( 1 ) = 4 x 3 e y + 2 x y 4 + 0 = 4 x 3 e y + 2 x y 4 Again, obtain the partial derivative of the function f x ( x , y ) with respect to x . f x x ( x , y ) = ∂ ∂ x ( 4 x 3 e y + 2 x y 4 ) = ∂ ∂ x ( 4 x 3 e y ) + ∂ ∂ x ( 2 x y 4 ) = 12 x 2 e y + 2 y 4 That is, f x x ( x , y ) = 12 x 2 e y + 2 y 4 . …… (1) Calculate the partial derivative of the function f ( x , y ) with respect to y . f y ( x , y ) = ∂ ∂ y ( x 4 e y + x 2 y 4 + 1 ) = ∂ ∂ y ( x 4 e y ) + ∂ ∂ y ( x 2 y 4 ) + ∂ ∂ y ( 1 ) = x 4 e y + x 2 ( 4 y 3 ) + 0 = x 4 e y + 4 x 2 y 3 Again, obtain the partial derivative of the function f y ( x , y ) with respect to y . f y y ( x , y ) = ∂ ∂ y ( f y ( x , y ) ) = ∂ ∂ y ( x 4 e y + 4 x 2 y 3 ) = ∂ ∂ y ( x 4 e y ) + ∂ ∂ y ( 4 x 2 y 3 ) = x 4 e y + 12 x 2 y 2 That is, f y y ( x , y ) = x 4 e y + 12 x 2 y 2 . …… (2) Calculate the partial derivative of the function f x ( x , y ) with respect to y . f x y ( x , y ) = ∂ ∂ y ( f x ( x , y ) ) = ∂ ∂ y ( 4 x 3 e y + 2 x y 4 ) = ∂ ∂ y ( 4 x 3 e y ) + ∂ ∂ y ( 2 x y 4 ) = 4 x 3 e y + 8 x y 3 That is, f x y ( x , y ) = 4 x 3 e y + 8 x y 3 (B) To determine To decide: Whether the function f ( x , y ) has a local maximum , a minimum, or saddle point at ( 0 , 0 ) .

Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

13th Edition

ISBN: 9780321869838

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Publisher: PEARSON

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Chapter 7.3, Problem 35E

(A)

To determine

To show: The function f(x,y)=x4ey+x2y4+1 has a critical point (0,0) and prove that second derivative test for local extrema fails at that point.

(B)

To determine

To decide: Whether the function f(x,y) has a local maximum, a minimum, or saddle point at (0,0).

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Chapter 7.3, Problem 34E

Chapter 7.3, Problem 36E

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Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

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The function f ( x , y ) = x 4 e y + x 2 y 4 + 1 has a critical point ( 0 , 0 ) and prove that second derivative test for local extrema fails at that point.

Solution Summary: The author demonstrates that the function f(x,y)=x4ey+ (0,0 ) has a critical point and proves