Lagrange Multiplier Calculator

The Lagrange multiplier method finds extrema of a function f(x,y) subject to a constraint g(x,y) = 0. The key insight: at an extremum on the constraint curve, the gradient of f must be parallel to the gradient of g. They cannot point in independent directions, or you could slide along the constraint and push f higher.

So the method solves: ∇f = λ ∇g, plus the constraint g = 0.

λ is not just bookkeeping. Most textbooks introduce λ as an auxiliary unknown you solve for and discard, and that undersells it. λ measures how much the optimum would improve if you loosened the constraint by one unit. Economists call it the shadow price. Write the circle constraint as x² + y² = c and the maximum works out to √c · √(a² + b²); differentiate that with respect to c and you get exactly √(a² + b²) / (2r), which is λ. So on the worked example below, λ = 0.5 says that raising c from 25 to 26 would buy you about half a unit of extra objective. If your constraint is a budget, λ is what one more dollar is worth.

This calculator handles two standard constraint shapes with a linear objective f(x,y) = ax + by.

Circle constraint x² + y² = r²: ∇f = (a, b), ∇g = (2x, 2y). Setting (a,b) = λ(2x, 2y) gives x = a/(2λ), y = b/(2λ). Substituting into the constraint: (a² + b²)/(4λ²) = r², so 2λ = ±√(a² + b²)/r.

Critical points: ±r · (a, b) / √(a² + b²). The maximum is f_max = +r · √(a² + b²); the minimum is the negative.

Ellipse constraint x²/p² + y²/q² = 1: The same setup but with weighted gradient ∇g = (2x/p², 2y/q²). Critical points have x = ap²/(2λ), y = bq²/(2λ). Constraint substitution gives 2λ = ±√(a²p² + b²q²). Critical points: ±(ap², bq²)/√(a²p² + b²q²). The extremum values are ±√(a²p² + b²q²).

Worked example: maximize 3x + 4y on the circle of radius 5. f_max = 5 × √(9+16) = 5 × 5 = 25, achieved at (3, 4). Minimum is -25 at (-3, -4). Sanity check: the gradient (3,4) is exactly the position vector at the maximum point, so they are parallel as required.

The geometric picture: the level curves of f = ax + by are parallel lines. The maximum on the constraint curve is the line farthest in the gradient direction that still touches the curve, and that touchpoint is exactly where ∇f and ∇g are parallel.

Lagrange multipliers extend to multiple constraints (one λ per constraint) and to nonlinear objectives. The general principle remains: at any constrained extremum, the gradient of f lies in the span of the constraint gradients.