Explanation Theorem 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 , f y ( a , b ) = 0 , then (a) The function has relative 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 relative 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.” Calculation: Compute the partial derivative of the function f ( x , y ) = 100 + 6 x y − 4 x 2 − 3 y 2 with respect to x . f x ( x , y ) = ∂ ∂ x ( 100 + 6 x y − 4 x 2 − 3 y 2 ) = ∂ ∂ x ( 100 ) + 6 ∂ ∂ x ( x y ) − 4 ∂ ∂ x ( x 2 ) − 3 ∂ ∂ x ( y 2 ) = 0 + 6 y − 8 x − 0 = − 8 x + 6 y That is, f x ( x , y ) = − 8 x + 6 y . Again, compute the partial derivative of the function f x ( x , y ) with respect to x . f x x ( x , y ) = ∂ ∂ x ( f x ( x , y ) ) = ∂ ∂ x ( − 8 x + 6 y ) = − ∂ ∂ x ( 8 x ) + 6 ∂ ∂ x ( y ) = − 8 That is, f x x ( x , y ) = − 8 . …… (1) Compute the partial derivative of the function f ( x , y ) with respect to y . f y ( x , y ) = ∂ ∂ y ( 100 + 6 x y − 4 x 2 − 3 y 2 ) = ∂ ∂ y ( 100 ) + ∂ ∂ y ( 6 x y ) − 4 ∂ ∂ y ( x 2 ) − 3 ∂ ∂ y ( y 2 ) = 0 + 6 x − 0 − 6 y = 6 x − 6 y That is, f y ( x , y ) = 6 x − 6 y . Again, compute 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 ( 6 x − 6 y ) = 6 ∂ ∂ y ( x ) − 6 ∂ ∂ y ( y ) = − 6 That is, f y y ( x , y ) = − 6

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137553341

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

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Chapter 14.3, Problem 23E

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To find: The local extrema of the function f(x,y)=100+6xy−4x2−3y2.

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Chapter 14.3, Problem 22E

Chapter 14.3, Problem 24E

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The local extrema of the function f ( x , y ) = 100 + 6 x y − 4 x 2 − 3 y 2 .

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To find: The local extrema of the function f(x,y)=100+6xy−4x2−3y2.

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