--- title: "Lagrange Multipliers" description: "Optimise subject to a constraint by setting the gradient of the objective parallel to the gradient of the constraint, which is the moment the level curves touch." canonical: https://lightmysky.com/learn/mathematics/lagrange-multipliers-mt_0RPOBWZ1gs source: https://lightmysky.com/learn/mathematics/lagrange-multipliers-mt_0RPOBWZ1gs.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Lagrange Multipliers Optimise subject to a constraint by setting the gradient of the objective parallel to the gradient of the constraint, which is the moment the level curves touch. Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/lagrange-multipliers-mt_0RPOBWZ1gs ## Ready when they can - Set up the multiplier equations for a constrained problem - Explain the tangency condition in terms of level curves - Compare the multiplier method with substituting the constraint directly ## Lesson: Best point on a fixed curve Optimising with a constraint means staying on a curve while pushing the objective as far as possible. The best feasible point is where a level curve of f just touches the constraint instead of crossing it. Touching curves share a tangent, so grad f = lambda grad g, solved together with the constraint itself. **Example.** A rectangle with perimeter 20 obeys x + y = 10. The multiplier equations give x = y, so the winner is a 5 by 5 square with area 25. Likewise, x + y on x squared + y squared = 2 peaks at x = y = 1 with value 2. Lambda is not a spare part; it is the exchange rate. It says how much the optimum improves per unit of relaxed constraint. When substitution is easy both routes agree, but on a circle isolating a variable creates square roots that are painful to differentiate. **Tip.** The method finds suspects; the comparison convicts. Evaluate f at every candidate, then keep the largest or smallest, and check boundary behaviour plus any nonsmooth points. For x squared + y squared with x + y = 4, symmetry picks (2, 2) and the minimum 8. **Recap.** Set the gradients parallel, add the constraint, solve, and compare candidates before declaring a winner. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Critical Points and Optimisation in Two Variables](https://lightmysky.com/learn/mathematics/critical-points-and-optimisation-in-two-variables-mt_4qB5pAGeuR) - [Directional Derivatives and the Gradient](https://lightmysky.com/learn/mathematics/directional-derivatives-and-the-gradient-mt_LW_KckY5Ad)