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”). It’s a useful technique, but all too often it is poorly taught and poorly understood. With luck, this overview will help to make the concept and its applications a bit clearer.
Be warned: this page may not be what you’re looking for! If you’re looking for detailed proofs, I recommend consulting your favorite textbook on multivariable calculus: my focus here is on concepts, not mechanics. (Comes of being a physicist rather than a mathematician, I guess.) If you want to know about Lagrange multipliers in the calculus of variations, as often used in Lagrangian mechanics in physics, this page only discusses them briefly.
Here’s a basic outline of this discussion:
The meaning of the multiplier (inspired by physics and economics)
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