Multivariable optimisation and Lagrange multipliers

Finding where a multivariable function is largest or smallest starts the same way single-variable optimisation does: a smooth local extremum can only sit where every partial derivative is zero, so ∇f = 0 gives a set of candidate points. Unlike the single-variable case, a flat gradient no longer separates neatly into "max" or "min" — it might be a saddle point, flat in one direction and curved oppositely in another. The Hessian (the matrix of second partial derivatives) settles this: its sign pattern distinguishes a genuine maximum, a genuine minimum, and a saddle.

Constrained optimisation changes the question: instead of the best point anywhere in the plane, you want the best point on a curve g(x, y) = c. At that constrained extremum, the level curve of f is tangent to the constraint curve — geometrically, if it weren't tangent, you could slide slightly along the constraint and still improve f. Tangency means the two gradients are parallel, which is exactly the equation ∇f = λ∇g. That equation is not the whole problem, though: solving it only pins down a direction relative to the constraint, so the constraint equation g(x, y) = c must be solved alongside it to get an actual point.

The multiplier λ is not disposable bookkeeping — it measures the sensitivity of the optimal value of f to a small change in the constraint level c, the same interpretation that gives Lagrange multipliers their central role in economics (shadow prices) and physics (constraint forces).