Lagrange Multiplier Method

In an unconstrained optimization problem we set the gradient of a function equal to zero. The Lagrangian transforms the constrained optimization problem into an unconstrained optimization problem. 
Since \lambda is the ratio between the gradients it shows how fast the optimal value changes as the (budget) constraint is relaxed; \lambda = {\partial f^{max} \over \partial B}.
If the gradient of the objective function and the gradient of the constraint function are proportional at a certain point, then this point is an optimal point.