Consider minimizing $f(x)$ subject to an equality constraint

$$g(x)=0.$$

At a regular constrained optimum, motion tangent to the constraint cannot decrease the objective. The gradients of the objective and constraint must therefore be aligned:

$$\nabla f(x_)+\lambda\nabla g(x_)=0.$$

The scalar $\lambda$ is a Lagrange multiplier.

Define the Lagrangian

$$\mathcal L(x,\lambda)=f(x)+\lambda g(x).$$

Candidate constrained optima satisfy

$$\nabla_x\mathcal L=0,\qquad g(x)=0.$$

With several equality constraints, each receives its own multiplier.

Lagrange multipliers convert the geometric requirement of zero feasible first-order change into a system of equations. They identify candidates; additional analysis is needed to determine which candidates are actual minima or maxima.