Lagrange Multipliers

The central idea

To optimize differentiable $f:\mathbb R^n\to\mathbb R$ subject to a regular constraint $g(\mathbf x)=c$, candidate constrained extrema satisfy $\nabla f=\lambda\nabla g$ with $g=c$, provided $\nabla g\ne0$. With several independent constraints, add one multiplier per constraint.

Definitions, hypotheses, and notation

The regularity condition is essential: if $\nabla g=0$, the constraint may not have a smooth tangent space and the multiplier equation can miss or overproduce candidates. Constraints with endpoints, corners, or inequalities require boundary decomposition or Karush–Kuhn–Tucker ideas beyond the basic theorem.

For a compact constraint and continuous objective, absolute extrema exist by the Extreme Value Theorem. Lagrange multipliers then generate candidates, but global classification comes only from evaluating all candidates. The sign and size of $\lambda$ should not be mistaken for max/min classification.

Conceptual meaning

At a constrained extremum, every feasible tangent direction has zero directional change in $f$. Thus $\nabla f$ is normal to the constraint surface, as is $\nabla g$, making them parallel. The multiplier records sensitivity to the constraint level under suitable conditions.

A dependable method and decision rule

1. Check compactness or other conditions if absolute extrema are requested.
2. Compute both gradients and note singular constraint points separately.
3. Solve $\nabla f=\lambda\nabla g$ together with the constraint.
4. Include all solution branches, especially zero-coordinate cases.
5. Evaluate $f$ at every candidate and compare values.

Fully worked example

Common mistakes and why they fail