Solve constrained extrema for f(x,y) subject to constraint h(x,y)=0 using the Lagrange multiplier method:
∇f = λ∇h and h=0. You may enter the constraint as an equation like
x^2 - y^2 = 6 — it will be rewritten to zero-form.
=, only the left side is used.a=b are auto-rewritten to (a)-(b).(x,y,λ) with
fₓ = λhₓ, fᵧ = λhᵧ, and h=0.
x^2 - y^2 = 6 are converted to x^2 - y^2 - 6.= too (left side is used).sin, cos, exp, log, sqrt.A Lagrange Multiplier Calculator is a tool for optimizing an objective function subject to one or more equality constraints. It automates the classical method of introducing multipliers for each constraint, building the first-order system, and solving for stationary points that satisfy feasibility. The calculator reports candidate optima and, when requested, applies second-order tests (e.g., bordered Hessian) to classify maxima, minima, or saddle points. All equations are displayed in LaTeX and render responsively with MathJax or math.js.
Given an objective \(f(\mathbf{x})\) with constraints \(g_i(\mathbf{x})=0\) for \(i=1,\dots,m\), the method searches for points where the gradient of \(f\) lies in the span of constraint gradients. This converts a constrained problem into solving simultaneous equations in \(\mathbf{x}\) and multipliers \(\lambda_i\). The tool supports symbolic entry of \(f\) and \(g_i\), numeric solving, multiple constraints, and optional inequality handling via KKT hints. It also checks constraint qualification issues (e.g., linearly dependent gradients) and warns when solutions may be degenerate or non-isolated.
First-order conditions (m constraints): \[ \nabla f(\mathbf{x}^\star) = \sum_{i=1}^{m} \lambda_i \nabla g_i(\mathbf{x}^\star), \qquad g_i(\mathbf{x}^\star)=0 \ (i=1,\dots,m). \]
Single-constraint system: \[ \begin{cases} \nabla f(\mathbf{x})=\lambda \nabla g(\mathbf{x})\\ g(\mathbf{x})=0 \end{cases} \]
Bordered Hessian (classification, one constraint): \[ H_B= \begin{bmatrix} 0 & \nabla g^\top\\ \nabla g & \nabla^2 f \end{bmatrix}, \quad \text{use principal minors’ signs to test max/min on the constraint manifold.} \]
KKT (inequality \(h_j(\mathbf{x})\le 0\)): \[ \nabla f=\sum_i \lambda_i \nabla g_i+\sum_j \mu_j \nabla h_j,\; g_i=0,\; h_j\le0,\; \mu_j\ge0,\; \mu_j h_j=0. \]
Constraint \(g(x,y)=x^2+y^2-1=0\). Conditions: \[ \begin{cases} y=2\lambda x\\ x=2\lambda y\\ x^2+y^2=1 \end{cases} \Rightarrow \lambda=\pm \tfrac{1}{2}. \] For \(\lambda=\tfrac12\): \(y=x\Rightarrow (x,y)=\big(\tfrac{1}{\sqrt2},\tfrac{1}{\sqrt2}\big)\) gives maximum \(xy=\tfrac12\). For \(\lambda=-\tfrac12\): \(y=-x\Rightarrow (x,y)=\big(\tfrac{1}{\sqrt2},-\tfrac{1}{\sqrt2}\big)\) gives minimum \(xy=-\tfrac12\).
Minimize \(f(x,y)=x^2+y^2\) subject to \(g(x,y)=x+y-1=0\). \[ 2x=\lambda,\; 2y=\lambda,\; x+y=1 \Rightarrow x=y=\tfrac12,\ f=\tfrac12. \]
Maximize \(u(x,y)=x^{1/2}y^{1/2}\) subject to \(ax+by=m\). \[ \frac{\partial u}{\partial x}=\frac{1}{2}\frac{y^{1/2}}{x^{1/2}}=\lambda a,\quad \frac{\partial u}{\partial y}=\frac{1}{2}\frac{x^{1/2}}{y^{1/2}}=\lambda b \] \(\Rightarrow ax=by\) and \(ax+by=m \Rightarrow x=\tfrac{m}{2a},\; y=\tfrac{m}{2b}.\)
A scalar (or set of scalars) that balances the gradient of the objective against constraint gradients at an optimum.
Use them for smooth optimization with equality constraints when interior stationary points might exist.
Yes. It forms \(\nabla f=\sum_i \lambda_i \nabla g_i\) with all \(g_i(\mathbf{x})=0\) and solves the resulting system.
Use the KKT option: add multipliers \(\mu_j\ge0\) and complementary slackness \(\mu_j h_j=0\).
Enable classification to inspect the bordered Hessian or evaluate \(f\) around the feasible manifold.
The tool flags rank deficiency; constraint qualification may fail, so results require extra care.
Symbolic solves may not; numeric solvers benefit from reasonable starting points for faster convergence.
Yes. Stationary feasible points can be maxima, minima, or saddles; classification distinguishes them.
No. Solutions for \(\mathbf{x}\) are invariant; only multipliers rescale accordingly.