Explanation Given: The function is given as: z = x 3 + y 2 + 2 x y − 4 x − 3 y − 2 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 3 + y 2 + 2 x y − 4 x − 3 y − 2 with respect to x and take y as constant, by the use of the Power rule. For f ( x , y ) = x n + y n + x m y m 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 = 3 x 3 − 1 + 0 + 2 x 1 − 1 y − 4 x 1 − 1 − 0 − 0 = 3 x 2 + 2 y − 4 And f y = 0 + 2 y 2 − 1 + 2 x y 1 − 1 − 0 − 3 y 1 − 1 − 0 = 2 y + 2 x − 3 . Set each of these two partial derivatives equal to 0. 3 x 2 + 2 y − 4 = 0 and 2 y + 2 x − 3 = 0 . These two equations form a system of linear equations that can be rewritten as: 3 x 2 + 2 y − 4 = 0 2 y + 2 x − 3 = 0 To solve this system by elimination, multiply the second equation by ( − 1 ) and then add the equations: 3 x 2 + 2 y − 4 = 0 − 2 y − 2 x + 3 = 0 After addition, the equations are reduced to the form: 3 x 2 − 2 x − 1 = 0 3 x 2 − 3 x + x − 1 = 0 ( 3 x + 1 ) ( x − 1 ) = 0 x = − 1 3 , 1 Substitute the value of x as x = − 1 3 in the first equation of the system. 3 x 2 + 2 y − 4 = 0 3 ( − 1 3 ) 2 + 2 y − 4 = 0 1 3 + 2 y − 4 = 0 y = 11 6 Substitute x = 1 in the first equation of the system. 3 x 2 + 2 y − 4 = 0 3 ( 1 ) 2 + 2 y − 4 = 0 3 + 2 y − 4 = 0 y = 1 2 Thus, ( 1 , 1 2 ) , ( − 1 3 , 11 6 ) are the points where the given function z = x 3 + y 2 + 2 x y − 4 x − 3 y − 2 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 . Consider 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 . 2. There exists a local minimum at the point ( a , b ) if M > 0 and f x x ( a , b ) > 0 . 3. The saddle point is at the point ( a , b ) if M < 0 . 4. There is no information available if the value of M is M = 0 . To find f x x ( x , y ) differentiate f x with respect to x and take y as constant, use the power rule for f x ( x , y ) = n x n − 1 + m x m − 1 y m

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ISBN: 9780135240687

Author: Lial

Publisher: PEARSON

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Chapter 14, Problem 43RE

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To calculate: All the points where the provided function z=x3+y2+2xy−4x−3y−2 has any local extrema and saddle points.

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Chapter 14, Problem 42RE

Chapter 14, Problem 44RE

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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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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 z = x 3 + y 2 + 2 x y − 4 x − 3 y − 2 has any local extrema and saddle points.

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To calculate: All the points where the provided function z=x3+y2+2xy−4x−3y−2 has any local extrema and saddle points.

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Chapter 14 Solutions