lagrange.mult - Lagrange Multipliers Jonathan L.F King...

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Unformatted text preview: Lagrange Multipliers Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 9 November, 2011 (at 23:44 ) Cubic-surface Problem On R 3 , find all extreme points P = ( x,y,z ) of function ψ ( P ) := x 2 + y 2 + z 2 , subject to the two constraints that x · y · z = 54 ; ( a cubic surface ) C f : x + y + z = 12 . ( a plane ) C g : A picture. Note that the objective fnc and the two constraints are symmetric under all permuta- tions of x,y,z . Thus our set of Lagrange points must have a six-fold ( 6 = 3! ) symmetry. A good picture suggests that the only Lagrange pts have some two of x,y,z equal. In the case that x = z , say, a picture suggests that there are 3 so- lutions; two solns where the common value is pos- itive and one soln with it negative. Thus the z = y , y = x and x = z solutions would give us 9 Lagrange- pts all together, but only 3 types of Lagrange-pt. A solution Define specifier functions f ( P ) := x · y · z and g ( P ) := x + y + z , so that locus ( C f ) is some level-set of f , and lo- cus ( C g ) is some level-set of g ....
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lagrange.mult - Lagrange Multipliers Jonathan L.F King...

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