The method of Lagrange multipliers states that the extreme values of the objective function f ( x , y ) {\displaystyle f(x,y)} subject to the constraint g ( x , y ) = 0 {\displaystyle g(x,y)=0} will occur at the solutions to the system: ∇ f = ⟨ f x , f y ⟩ = λ ⟨ g x , g y ⟩ = λ ∇ g {\displaystyle \nabla f=\langle f_{x},f_{y}\rangle =\lambda \langle g_{x},g_{y}\rangle =\lambda \nabla g} .
