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# 10.17 - Section 11.8 Lagrange multipliers(concluded On to...

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Section 11.8: Lagrange multipliers (concluded) On to three dimensions! Suppose ( x 0 , y 0 , z 0 ) is a point on the surface g ( x , y , z ) = k at which the function f ( x , y , z ) is maximized or minimized, with f , g differentiable. Then f ( x 0 , y 0 , z 0 ) and g ( x 0 , y 0 , z 0 ) are parallel. So as long as g ( x 0 , y 0 , z 0 ) 0 , we can write f ( x 0 , y 0 , z 0 ) = λ g ( x 0 , y 0 , z 0 ). Traditional method: Write f = λ g (but then we have to worry separately about the points where g = 0 ). Modern method: Write f × g = 0 . Example: Find the minimum of f ( x , y , z ) = x 2 + y 2 + z 2 on the set of points ( x , y , z ) with x + 2 y + 2 z = 18. Traditional way: Write 2 x ,2 y ,2 z = λ 1,2,2 , x + 2 y + 2 z = 18. x = λ /2, y = λ , z = λ λ /2 + 2 λ + 2 λ = 18 λ = 4 ( x , y , z ) = (2,4,4) Modern way: 0 = f × g = 2 x ,2 y ,2 z × 1,2,2 = 4 y – 4 z , 2 z – 4 x , 4 x – 2 y and x + 2 y + 2 z = 18;

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4 y – 4 z = 0, 2 z – 4 x = 0, 4 x – 2 y = 0, x + 2 y + 2 z = 18; y = z = 2 x , x + 4 x + 4 x = 18, x = 2, etc. Optimization under two simultaneous constraints: Suppose ( x 0 , y 0 , z 0 ) is a point on the curve C that is the intersection of the levels sets g
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10.17 - Section 11.8 Lagrange multipliers(concluded On to...

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