A method for maximizing or minimizing a general function f(x,y,z) subject
to a constraint (or side condition) of the form g(x,y,z)=k.
Theorem
Suppose f,g∈C1 and ∇g=0. Then the extremum(s) of f(x,y) subjected to g(x,y)=k are included in each of:
det∂(x,y)∂(f,g)=0andg(x,y)=0
∇f(x,y)=λ∇g(x,y)andg(x,y)=0
When there are n constraint, the system of equations becomes:
The system of equations is⎩⎨⎧∇f=λ1∇g1+λ2∇g2+⋯+λn∇gng1(x,y)=k1g2(x,y)=k2⋮gn(x,y)=kn