LaTeX
\nabla f(x)=\lambda\nabla g(x)Variables
- f: objective
- g(x)=c: constraint
- λ: Lagrange multiplier
How to use this formula
Gives a first-order condition for a constrained extremum with one equality constraint.
Important notes
- Constraint qualifications and second-order checks are still required.
Quick example
Optimize area subject to a fixed perimeter by solving the gradient equations.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Constraint qualifications and second-order checks are still required.
- Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result.
Worked example
Optimize area subject to a fixed perimeter by solving the gradient equations.
Common mistakes
- Before substituting values into Single-Constraint Lagrange Condition, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions.
Continue the workflow
Use Single-Constraint Lagrange Condition in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\nabla f(x)=\lambda\nabla g(x). - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Single-Constraint Lagrange Condition used for?
Gives a first-order condition for a constrained extremum with one equality constraint.
Can I copy this formula as LaTeX?
Yes. Copy \nabla f(x)=\lambda\nabla g(x) or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.