Explanation Given: The function is given as: f ( x , y ) = 7 x 2 + y 2 − 5 x y − 3 x + 6 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 . To find f x ( x , y ) differentiate f ( x , y ) = 7 x 2 + y 2 − 5 x y − 3 x + 6 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 . The value of f x is obtained as: 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 = 7 ( 2 ) x 2 − 1 + 0 − 5 x 1 − 1 y − 3 x 1 − 1 + 0 = 14 x − 5 y − 3 And, f y = 0 + 2 y 2 − 1 − 5 x y 1 − 1 − 0 + 6 y 1 − 1 = 2 y − 5 x + 6 . Set each of these two partial derivatives equal to 0: 14 x − 5 y − 3 = 0 and 2 y − 5 x + 6 = 0 . These two equations form a system of linear equations as: 14 x − 5 y − 3 = 0 2 y − 5 x + 6 = 0 To solve this system by elimination, multiply the first equation by 2, second equation by 5 and then add the equations: 28 x − 10 y − 6 = 0 10 y − 25 x + 30 = 0 After addition, we obtain 28 x − 25 x − 6 + 30 = 0 3 x = − 24 x = ( − 8 ) Substitute x = − 8 in the first equation of the system 14 x − 5 y − 3 = 0 14 ( − 8 ) − 5 y − 3 = 0 − 5 y = 115 y = ( − 23 ) Thus, ( − 8 , − 23 ) are the points where the given function f ( x , y ) = 7 x 2 + y 2 − 5 x y − 3 x + 6 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 . Assume M be the number defined by: M = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 1. There exists a local maximum at the point ( a , b ) if the value of M is M > 0 and f x x ( a , b ) < 0

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ISBN: 8220101336323

Author: MULLINS

Publisher: PEARSON

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Chapter 14, Problem 44RE

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To calculate: All the points where the provided function f(x,y)=7x2+y2−5xy−3x+6y has any local extrema and saddle points.

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Chapter 14, Problem 43RE

Chapter 14, Problem 45RE

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Checkout Lines According to...Ch. 14.1 - Prob. 40E Ch. 14.1 - Prob. 41E Ch. 14.1 - Prob. 42E Ch. 14.1 - Prob. 43E Ch. 14.1 - Prob. 44E Ch. 14.1 - Prob. 45E Ch. 14.1 - Prob. 46E Ch. 14.1 - Prob. 47E Ch. 14.1 - Prob. 48E Ch. 14.2 - Prob. 1CP Ch. 14.2 - Prob. 2CP Ch. 14.2 - Prob. 3CP Ch. 14.2 - Prob. 4CP Ch. 14.2 - Prob. 5CP Ch. 14.2 - Prob. 6CP Ch. 14.2 - Prob. 7CP Ch. 14.2 - Prob. 8CP Ch. 14.2 - Prob. 9CP Ch. 14.2 - Prob. 1E Ch. 14.2 - For each of the given functions, find the...Ch. 14.2 - Prob. 3E Ch. 14.2 - Prob. 4E Ch. 14.2 - Prob. 5E Ch. 14.2 - Prob. 6E Ch. 14.2 - Prob. 7E Ch. 14.2 - Prob. 8E Ch. 14.2 - Prob. 9E Ch. 14.2 - Prob. 10E Ch. 14.2 - Prob. 11E Ch. 14.2 - Prob. 12E Ch. 14.2 - Prob. 13E Ch. 14.2 - Prob. 14E Ch. 14.2 - Prob. 15E Ch. 14.2 - Prob. 16E Ch. 14.2 - Prob. 17E Ch. 14.2 - Prob. 18E Ch. 14.2 - Prob. 19E Ch. 14.2 - Prob. 20E Ch. 14.2 - Prob. 21E Ch. 14.2 - Prob. 22E Ch. 14.2 - Prob. 23E Ch. 14.2 - Prob. 24E Ch. 14.2 - Prob. 25E Ch. 14.2 - Prob. 26E Ch. 14.2 - Prob. 27E Ch. 14.2 - Prob. 28E Ch. 14.2 - Prob. 29E Ch. 14.2 - Prob. 30E Ch. 14.2 - Prob. 31E Ch. 14.2 - Prob. 32E Ch. 14.2 - Prob. 33E Ch. 14.2 - Prob. 34E Ch. 14.2 - Prob. 35E Ch. 14.2 - Prob. 36E Ch. 14.2 - Prob. 37E Ch. 14.2 - Find fx,fy,fz, and fyz for the given functions. 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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All the points where the provided function f ( x , y ) = 7 x 2 + y 2 − 5 x y − 3 x + 6 y has any local extrema and saddle points.

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To calculate: All the points where the provided function f(x,y)=7x2+y2−5xy−3x+6y has any local extrema and saddle points.

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Chapter 14 Solutions