Mr. Integral
Math notes/Calculus III/Optimization in Several Variables

Lagrange Multipliers and Constraint Geometry — Cheat sheet

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Core rule

∇f=λ∇g,g=c,∇g≠0.\nabla f=\lambda\nabla g,\quad g=c,\quad \nabla g\ne0.

Watch for

Compare candidates and check singular constraint points. Multiple constraints require independent gradients for the usual theorem.