= Solution
Assume $f$ and $g$ are continuously differentiable near the point. At a regular <feasible point>, meaning $g(x)=0$ and $\nabla g(x)\ne0$, a differentiable constrained local extremum satisfies the <Lagrange multiplier> equations
$$
\nabla f(x)=\lambda\nabla g(x),\qquad g(x)=0.
$$
They arise by differentiating the <optimization Lagrangian> $L(x,\lambda)=f(x)-\lambda g(x)$ with respect to all coordinates and the multiplier. Geometrically, the objective has zero <derivative> along the <tangent space> of the <constraint> surface, so its <gradient> must lie in the normal direction.
Solve these equations for candidates, examine <feasible points> where the regularity condition fails separately, and compare objective values or use a second-order test restricted to feasible directions. The equations locate <stationary points>; they do not by themselves establish which are minima or whether a <global minimum> is attained.
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