Explanation Given Information: The provided function is, g ( x , y ) = x 2 + y 2 + 2 x y Formula used: If ( a , b ) is a critical point in the interior of domain of any function f of two variables, then let H be a mathematical quantity, H = f x x ( a , b ) f y y ( a , b ) − [ f x y ( a , b ) ] 2 If, H is positive, then f has relative minimum at ( a , b ) if f x x ( a , b ) > 0 and f has relative maximum at ( a , b ) if, f x x ( a , b ) < 0 . And, if H is negative, then f has a saddle point at ( a , b ) . Calculation: Consider the provided function, g ( x , y ) = x 2 + y 2 + 2 x y Now, obtain the first order partial derivatives of this function as, g x = 2 x − 2 x 2 y And, g y = 2 y − 2 x y 2 Obtain the critical points by setting g x and g y equal to zero as, g x = 0 2 x = 2 x 2 y y = 1 x 3 And, g y = 0 2 y = 2 x y 2 x = 1 y 3 Substitute 1 x 3 for y in above equation to get, x = 1 ( 1 x 3 ) 3 x = x 9 x 8 = 1 x = ± 1 Because, x = 0 is rejected as it cannot be in the domain of the function f . Thus, y = 1 x 3 = 1 ± 1 = ± 1 Thus, the critical points are ( 1 , 1 ) and ( − 1 , − 1 )

Student Solutions Manual for Waner/Costenoble's Applied Calculus, 7th

7th Edition

ISBN: 9781337291293

Author: Waner, Stefan; Costenoble, Steven

Publisher: Brooks Cole

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Chapter 8.3, Problem 35E

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To calculate: The critical points for the function g(x,y)=x2+y2+2xy.

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Chapter 8.3, Problem 34E

Chapter 8.3, Problem 36E

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Chapter 8 Solutions

Student Solutions Manual for Waner/Costenoble's Applied Calculus, 7th

Ch. 8.1 - For each function in Exercises 14, evaluate (a)...Ch. 8.1 - For each function in Exercises 14, evaluate (a)...Ch. 8.1 - For each function in Exercises 14, evaluate (a)...Ch. 8.1 - For each function in Exercises 14, evaluate (a)...Ch. 8.1 - For each function in Exercises 5-8, evaluate (a)...Ch. 8.1 - Prob. 6E Ch. 8.1 - For each function in Exercises 5-8, evaluate (a)...Ch. 8.1 - For each function in Exercises 5-8, evaluate (a)...Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

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The critical points for the function g ( x , y ) = x 2 + y 2 + 2 x y .

Solution Summary: The author calculates the critical points of the function g(x,y)=x2+y

To calculate: The critical points for the function g(x,y)=x2+y2+2xy.

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Chapter 8 Solutions