Explanation 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 and the discriminant D = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 . (a) The function has relative maximum at ( a , b ) if D > 0 and f x x ( a , b ) < 0 . (b) The function has relative minimum at ( a , b ) if D > 0 and f x x ( a , b ) > 0 . (c) If D < 0 , then f ( a , b ) is a saddle point (neither a maximum nor a minimum) (d) If D = 0 , then the test fail. Calculation: The given function is, f ( x , y ) = x 2 + x y + y 2 − 6 x − 3 . Calculate the partial derivative of the above function with respect to x . f x ( x , y ) = ∂ ∂ x ( x 2 + x y + y 2 − 6 x − 3 ) = ∂ ∂ x ( x 2 ) + ∂ ∂ x ( x y ) + ∂ ∂ x ( y 2 ) − ∂ ∂ x ( 6 x ) − ∂ ∂ x ( 3 ) = 2 x + y + 0 − 6 − 0 = 2 x + y − 6 Again, take 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 ( 2 x + y − 6 ) = ∂ ∂ x ( 2 x ) + ∂ ∂ x ( y ) − ∂ ∂ x ( 6 ) = 2 − 0 + 0 = 2 That is, f x x ( x , y ) = 2 (1) Calculate the partial derivative of the function f ( x , y ) with respect to y . f y ( x , y ) = ∂ ∂ y ( x 2 + x y + y 2 − 6 x − 3 ) = ∂ ∂ y ( x 2 ) + ∂ ∂ y ( x y ) + ∂ ∂ y ( y 2 ) − ∂ ∂ y ( 6 x ) − ∂ ∂ y ( 3 ) = 0 + x + 2 y + 0 + 0 = x + 2 y Again, take the partial derivative of the function f y ( x , y ) with respect to y

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 17.3, Problem 4E

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To find: The points such that the function f(x,y)=x2+xy+y2−6x−3 attains relative extrema and identify the saddle points.

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Chapter 17.3, Problem 3E

Chapter 17.3, Problem 5E

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Chapter 17 Solutions

Finite Mathematics and Calculus with Applications

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Prob. 61E Ch. 17.2 - Prob. 62E Ch. 17.2 - Drug Reaction The reaction to x units of a drug t...Ch. 17.2 - Prob. 64E Ch. 17.2 - Prob. 66E Ch. 17.2 - Prob. 67E Ch. 17.2 - Prob. 68E Ch. 17.2 - Gravitational Attraction The gravitational...Ch. 17.2 - Velocity In 1931, Albert Einstein developed the...Ch. 17.2 - Prob. 71E Ch. 17.3 - Prob. 1YT Ch. 17.3 - Prob. 2YT Ch. 17.3 - Prob. 1WE Ch. 17.3 - Prob. 2WE Ch. 17.3 - Prob. 1E Ch. 17.3 - Prob. 2E Ch. 17.3 - Prob. 3E Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - Prob. 6E Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Prob. 9E Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - Prob. 16E Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.3 - Prob. 19E Ch. 17.3 - Prob. 20E Ch. 17.3 - Prob. 21E Ch. 17.3 - Prob. 22E Ch. 17.3 - Prob. 23E Ch. 17.3 - Figures (a)(f) show the graphs of the functions...Ch. 17.3 - Prob. 25E Ch. 17.3 - Prob. 26E Ch. 17.3 - Prob. 27E Ch. 17.3 - 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The points such that the function f ( x , y ) = x 2 + x y + y 2 − 6 x − 3 attains relative extrema and identify the saddle points.

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To find: The points such that the function f(x,y)=x2+xy+y2−6x−3 attains relative extrema and identify the saddle points.

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Chapter 17 Solutions