Box Optimization | Woody Calculus

Lagrange Multipliers Explained: Why Gradients Become Parallel

July 23, 2026 Brian M. Woody Leave a comment

Lagrange multiplier diagram showing level curves of f(x,y), constraint g(x,y)=c, and parallel gradients ∇f and ∇g at an optimum, from Woody Calculus.

Why do gradients become parallel in constrained optimization? This Woody Calculus lesson explains the geometry and algebra of Lagrange multipliers, derives the system ∇f = λ∇g, and solves complete rectangle and box optimization examples.

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