I'm trying to solve a Lagrangian multiplier equation however the partial derivatives seem to result in a contradiction (the last 4 equations). Any help is much appreciated! thank you in advance.
To maximize:
U(x,y,z) = 1000/(1+e^(-0.01(x-500) ) )+250lg(y)-100 sin²(0.01z)+1.2z
Subjected to
5x + 30y + 50z ≤ 1500
5x ≤ 600
30y ≤ 600
50z ≤ 600
The Lagrange function should result in:
U(x,y,z,λ_1,λ_2,λ_3,λ_4 ) = 1000/(1+e^(-0.01(x-500) ) )+250lg(y)-100 sin^2(0.01z)+1.2z+λ_1 (5x+30y+50z-1500)+λ_2 (5x-600)+λ_3 (30y-600)+λ_4 (50z-600)
∂U/∂x = (10e^(-0.01(x-500)))/((1+e^(-0.01(x-500) ) )^2 )+5λ_1+5λ_2 = 0
∂U/∂y = 250/(yln(10))+30λ_1+30λ_3 = 0
∂U/∂z = 1.2-sin(0.02z)+ 〖50λ〗_1+〖50λ〗_4 = 0
∂f/(∂λ_1 ) = 5x + 30y + 50z - 1500 = 0 (12)
∂f/(∂λ_2 ) = 5x - 600 = 0 (13)
∂f/(∂λ_3 ) = 30y - 600 = 0 (14)
∂f/(∂λ_4 ) = 50z - 600 = 0 (15)