Explanation Given Information: The provided function is z = x 2 − 4 x y + y 2 − 6 y . Formula used: Differentiate z = f ( x , y ) with respect to x by holding y constant as: ∂ ∂ x [ f ( x , y ) ] = f x ( x , y ) Differentiate f ( x , y ) with respect to y by holding x constant as: ∂ ∂ y [ f ( x , y ) ] = f y ( x , y ) The every differentiable function f ( x , y ) , ∂ 2 ∂ y ∂ x f ( x , y ) = ∂ 2 ∂ x ∂ y f ( x , y ) The following procedure are used to calculate relative maximum and saddle point of given function f ( x , y ) . Step-1: Calculate first derivative of function f ( x , y ) with respect to x and y . Step-2 Calculate second derivative of function f ( x , y ) with respect to x and y . Step-3: Equate first derivative of function f ( x , y ) with respect to x to zero. Step-4: Equate first derivative of function f ( x , y ) with respect to y to zero. Step-5: Calculate critical point of function f ( x , y ) that is ( a , b ) . Step-6: Now test function f ( x , y ) at critical point ( a , b ) for extrema and saddle point that is describe in following table. Critical point ∂ 2 ∂ 2 x f ( x , y ) ∂ 2 ∂ x 2 f ( x , y ) ∂ 2 ∂ y 2 f ( x , y ) − [ ∂ 2 ∂ y ∂ x f ( x , y ) ] 2 Conclusion ( a , b ) ∂ 2 ∂ 2 x f ( a , b ) > 0 ∂ 2 ∂ x 2 f ( a , b ) ∂ 2 ∂ y 2 f ( a , b ) − [ ∂ 2 ∂ y ∂ x f ( a , b ) ] 2 > 0 The function f ( x , y ) has relative minimum at point ( a , b ) ( a , b ) ∂ 2 ∂ 2 x f ( a , b ) < 0 ∂ 2 ∂ x 2 f ( a , b ) ∂ 2 ∂ y 2 f ( a , b ) − [ ∂ 2 ∂ y ∂ x f ( a , b ) ] 2 > 0 The function f ( x , y ) has relative maximum at point ( a , b ) ( a , b ) ∂ 2 ∂ x 2 f ( a , b ) ∂ 2 ∂ y 2 f ( a , b ) − [ ∂ 2 ∂ y ∂ x f ( a , b ) ] 2 < 0 The function f ( x , y ) has saddle point ( a , b , f ( a , b ) ) ( a , b ) ∂ 2 ∂ x 2 f ( a , b ) ∂ 2 ∂ y 2 f ( a , b ) − [ ∂ 2 ∂ y ∂ x f ( a , b ) ] 2 = 0 The test gives no information. The simple power rule of derivative: d d x ( x n ) = n x n − 1 The constant rule of derivative: d d x ( c ) = 0 The derivative of constant coefficient: d d x ( c f ( x ) ) = c f ′ ( x ) Calculation: Consider the function, z = x 2 − 4 x y + y 2 − 6 y . Differentiate z = f ( x , y ) with respect to x by holding y constant, ∂ z ∂ x = ∂ ∂ x ( x 2 − 4 x y + y 2 − 6 y ) = 2 x − 4 y Equate the first derivative of function z = x 2 − 4 x y + y 2 − 6 y with respect to x to zero

Mathematical Applications for the Management, Life, and Social Sciences

11th Edition

ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Question

Chapter 14.4, Problem 10E

To determine

To calculate: The relative maxima, relative minima, and saddle points of z=x2−4xy+y2−6y, if they exist.

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Chapter 14.4, Problem 9E

Chapter 14.4, Problem 11E

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