= Solution
At a regular constrained extremum, every <tangent vector> to the <level set> $g=A$ gives zero <directional derivative> of $f$. Thus $\nabla f$ is perpendicular to the tangent <plane>, and must be a scalar multiple of its nonzero <normal vector> $\nabla g$. The <Lagrange multiplier> equations are
$$
\boxed{\nabla f=\lambda\nabla g,\qquad g=A}.
$$
One solves these four equations for $x,y,z,\lambda$, evaluates $f$ at the candidates, and classifies or compares the values as required. Points where $\nabla g=0$, and any boundary or limiting cases of the feasible domain, require separate checks because the regularity argument does not cover them.
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