Why we use lagrange multipliers

Lagrange multipliers are used in multivariable calculus to find maxima and minima of a function subject to constraints (like "find the highest elevation along the given path" or "minimize the cost of materials for a box enclosing a given volume").

1. What does the Lagrange multiplier tell us?
2. What is Lagrangian used for?
3. When not to use Lagrange multipliers?

What does the Lagrange multiplier tell us?

The Lagrange multiplier, λ, measures the increase in the objective function (f(x, y) that is obtained through a marginal relaxation in the constraint (an increase in k). For this reason, the Lagrange multiplier is often termed a shadow price.

What is Lagrangian used for?

How a special function, called the "Lagrangian", can be used to package together all the steps needed to solve a constrained optimization problem.

When not to use Lagrange multipliers?

Recall that a minimum for a differentiable function occurs either at a point where the derivative is 0, or on the boundary. If the minimum is an interior point, the Lagrange multipliers won't matter.