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

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Unformatted text preview: Section 11.8: Lagrange multipliers (concluded) On to three dimensions! Suppose ( x , y , z ) 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 , y , z ) and ∇ g ( x , y , z ) are parallel. So as long as ∇ g ( x , y , z ) ≠ , we can write ∇ f ( x , y , z ) = λ ∇ g ( x , y , z ). Traditional method: Write ∇ f = λ ∇ g (but then we have to worry separately about the points where ∇ g = ). Modern method: Write ∇ f × ∇ g = . 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: = ∇ 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; 4 y – 4...
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This note was uploaded on 02/13/2012 for the course MATH 241 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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

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