Lagrange Multipliers Calculator
Verified Calculation Engine
The online Lagrange Multipliers Calculator helps you calculate instantly and solve problems related to Vector Calculus. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Engineering Math calculations.
Solve constrained optimization with Lagrange multipliers for two variables: maximize/minimize f(x,y) subject to g(x,y)=c. Finds stationary candidates, verifies ∇f=λ∇g and the constraint, and classifies maxima/minima.
∇f = λ ∇g, g(x,y)=c
Stationary points of f on the constraint satisfy the Lagrange equations together with g=c. Compare objective values across feasible candidates to classify maxima and minima. If ∇f is parallel to ∇g identically and f is constant on the constraint, every feasible point shares the same objective value.
- Two variables x,y; one equality constraint.
- Objective and constraint are polynomials in x,y (safe parser; no eval).
- Classification compares objective values across verified candidates.
Inputs
Results
Worked Examples
Max/min on a circle
f=x+y subject to x^2+y^2=2 → max f=2 at (1,1), min f=-2 at (-1,-1).
- objective: x+y
- constraint: x^2+y^2=2
- f_max: 2
- f_min: -2
Quadratic objective, linear constraint
f=x^2+y^2 subject to x+y=4 → unique minimum at (2,2), f=8.
- objective: x^2+y^2
- constraint: x+y=4
- f_min: 8
Multiple extrema (xy on unit circle)
f=xy subject to x^2+y^2=1 → extrema ±1/2 at the four (±1/√2, ±1/√2) sign patterns.
- objective: x*y
- constraint: x^2+y^2=1
- f_max: 0.5
- f_min: -0.5
About this calculator
Overview
The Lagrange Multipliers Calculator finds constrained extrema of a two-variable objective f(x,y) subject to an equality constraint g(x,y)=c by solving ∇f = λ∇g together with the constraint, then verifying every candidate.
When to use
Use for textbook constrained optimization, checking hand calculations of Lagrange systems, and distinguishing unique extrema from constant-objective-on-constraint cases.
Inputs explained
- Objective f(x,y) — polynomial expression in x and y
- Constraint g(x,y)=c — equality such as x^2+y^2=2 or x+y=4
Formula / method
Parse f and the constraint into bivariate polynomials (safe, no eval). Form gradients, eliminate λ via f_x g_y − f_y g_x = 0, solve with the constraint, verify ∇f=λ∇g and g=c, then classify by comparing objective values.
Worked example
For f=x+y and x^2+y^2=2, candidates (1,1) and (−1,−1) give f=2 (maximum) and f=−2 (minimum). For f=x^2+y^2 on x^2+y^2=4, the objective is constantly 4 on the constraint.
Interpreting results
Each reported candidate must satisfy both the Lagrange equations and the constraint. Maximum/minimum labels come from comparing feasible objective values, not from an arbitrary sign heuristic.
Assumptions
- Two variables and one equality constraint.
- Polynomial expressions in x,y.
- Floating-point residuals are checked within a small tolerance.
Limitations
- Not a general CAS for arbitrary transcendental constraints.
- Does not handle inequality constraints (KKT) on this page.
- Educational tool — verify critical designs independently.
Important note
Educational engineering-mathematics tool. Verify critical designs with hand analysis or certified software.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
Parse f and the constraint into bivariate polynomials (safe, no eval). Form gradients, eliminate λ via f_x g_y − f_y g_x = 0, solve with the constraint, verify ∇f=λ∇g and g=c, then classify by comparing objective values.
Example Calculation
For f=x+y and x^2+y^2=2, candidates (1,1) and (−1,−1) give f=2 (maximum) and f=−2 (minimum). For f=x^2+y^2 on x^2+y^2=4, the objective is constantly 4 on the constraint.
Frequently Asked Questions
What is Lagrange Multipliers Calculator?
- The Lagrange Multipliers Calculator finds constrained extrema of a two-variable objective f(x,y) subject to an equality constraint g(x,y)=c by solving ∇f = λ∇g together with the constraint, then verifying every candidate.
How does Lagrange Multipliers Calculator work?
- Parse f and the constraint into bivariate polynomials (safe, no eval). Form gradients, eliminate λ via f_x g_y − f_y g_x = 0, solve with the constraint, verify ∇f=λ∇g and g=c, then classify by comparing objective values.
Why use this Engineering Math calculator?
- Use for textbook constrained optimization, checking hand calculations of Lagrange systems, and distinguishing unique extrema from constant-objective-on-constraint cases.