Explanation Given: The function is given as: f ( x , y ) = x 2 + 3 y 2 + 5 x y − 10 x − 12 y Formula used: f x x ( x , y ) = ∂ f x ∂ x f y y ( x , y ) = ∂ f y ∂ y f x y ( x , y ) = ∂ f x ∂ y M = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 Calculation: Since the function is a polynomial function, so partial derivatives always exist. Find all the points ( a , b ) such that f x ( a , b ) = 0 and f y ( a , b ) = 0 . Assume z = f ( x , y ) . To find f x ( x , y ) differentiate f ( x , y ) = x 2 + 3 y 2 + 5 x y − 10 x − 12 y with respect to x and take y as constant, use the Power Rule for f ( x , y ) = x n + y n + x m y m . Calculate the value of f x . f x ( x , y ) = n x n − 1 + 0 + m x m − 1 y m = n x n − 1 + m x m − 1 y m Similarly, f y ( x , y ) = 0 + n y n − 1 + x m m y m − 1 = n y n − 1 + x m m y m − 1 Now, f x = 2 x 2 − 1 + 0 + 5 x 1 − 1 y − 10 x 1 − 1 − 0 = 2 x + 5 y − 10 And, f y = 0 + 3 ( 2 ) y 2 − 1 + 5 x y 1 − 1 − 0 − 12 y 1 − 1 = 6 y + 5 x − 12 . Set each of these two partial derivatives equal to 0. 2 x + 5 y − 10 = 0 And, 6 y + 5 x − 12 = 0 . These two equations form a system of linear equations that can be rewritten as: 2 x + 5 y − 10 = 0 6 y + 5 x − 12 = 0 To solve this system by elimination, multiply the first equation by 5, second equation by ( − 2 ) and then add the equations as follows: 10 x + 25 y − 50 = 0 − 12 y − 10 x + 24 = 0 After addition, 25 y − 12 y − 50 + 24 = 0 + 0 13 y − 26 = 0 13 y = 26 y = 2 Substitute y = 2 in the first equation of the system, 2 x + 5 ( 2 ) − 10 = 0 2 x = 0 x = 0 Thus, ( 0 , 2 ) are the points where the given function f ( x , y ) = x 2 + 3 y 2 + 5 x y − 10 x − 12 y have any local extrema. Suppose f ( x , y ) is a function such that f x x , f y y , and f x y all exist. Assume ( a , b ) be a critical point for which f x ( a , b ) = 0 and f y ( a , b ) = 0

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ISBN: 8220101336323

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Chapter 14, Problem 41RE

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To calculate: All the points where the provided function f(x,y)=x2+3y2+5xy−10x−12y has any local extrema and saddle points.

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Chapter 14, Problem 40RE

Chapter 14, Problem 42RE

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In...Ch. 14.2 - Prob. 39E Ch. 14.2 - Suppose z=f(x,y) describes the cost to build a...Ch. 14.2 - Prob. 41E Ch. 14.2 - 42. Business The revenue from the sale of x units...Ch. 14.2 - Prob. 62E Ch. 14.2 - Prob. 44E Ch. 14.2 - Prob. 45E Ch. 14.2 - Prob. 43E Ch. 14.2 - Prob. 42E Ch. 14.2 - Prob. 59E Ch. 14.2 - Prob. 50E Ch. 14.2 - Prob. 51E Ch. 14.2 - Work the following applied problems. (See Example...Ch. 14.2 - Prob. 53E Ch. 14.2 - Work the following marginal productivity problems....Ch. 14.2 - Prob. 55E Ch. 14.2 - Prob. 56E Ch. 14.2 - Prob. 58E Ch. 14.2 - Work the following applied problems. 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Find positive numbers x and y whose sum is 80...Ch. 14 - Prob. 52RE Ch. 14 - Prob. 53RE Ch. 14 - Prob. 54RE Ch. 14 - Prob. 1CE Ch. 14 - Prob. 2CE

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All the points where the provided function f ( x , y ) = x 2 + 3 y 2 + 5 x y − 10 x − 12 y has any local extrema and saddle points.

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To calculate: All the points where the provided function f(x,y)=x2+3y2+5xy−10x−12y has any local extrema and saddle points.

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Chapter 14 Solutions