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O (1) (13) O (9) O O (12) (10) (2) O (11) O O O (5) (4) (7) O O (3) O (6) O (8) O Problem Set #2 - Answer Key 1. The Lagrangian is: L:=x^2*y^3+lambda*(I-px*x-py*y); L := x 2 y 3 C λ I K px x K py y We take the two derivatives to get: dLdx:=diff(L,x); dLdx := 2 x y 3 K λ px dLdy:=diff(L,y); dLdy := 3 x 2 y 2 K λ py dLdlam:=diff(L,lambda); dLdlam := I K px x K py y Solving gives the solutions: solve({dLdx,dLdy,dLdlam},{x,y,lambda})[3]; x = 2 5 I px , λ = 108 625 1 py 3 px 2 , y = 3 5 I py 2, The Lagrangian is: L:=(x-2)*(y-3)+lambda*(I-px*x-py*y); L := x K 2 y K 3 C λ I K px x K py y dLdx:=diff(L,x); dLdx := y K 3 K λ px dLdy:=diff(L,y); dLdy := x K 2 K λ py dLdlam:=diff(L,lambda); dLdlam := I K px x K py y Solving gives the solutions: solve({dLdx,dLdy,dLdlam},{x,y,lambda}); λ = 1 2 I K 2 px K 3 py px py , y = 1 2 3 py C I K 2 px py , x = 1 2 2 px C I K 3 py px 3. The Lagrangian is: L:=2*ln(x)+3*ln(y)+lambda*(I-px*x-py*y); L := 2 ln x C 3 ln y C λ I K px x K py y dLdx:=diff(L,x); dLdx := 2 \$ 1 x K λ px dLdy:=diff(L,y);

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(18) (22) O (13) O O O O (17) O (19) O (24) (14) (16) O (15) O (23) (20) (21) O O O dLdy := 3 \$ 1 y K λ py dLdlam:=diff(L,lambda);
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