Lagrange multipliers with two constraints

The three-variable extension $\nabla f=\lambda\nabla g+\mu\nabla h$ , used when two constraint surfaces intersect in a curve.

When two constraints $g(x,y,z)=k_1$ and $h(x,y,z)=k_2$ are both imposed, each generically carves out a surface, and two generic surfaces meet along a curve — the one-dimensional constraint set the single-multiplier method was built for. The condition becomes $\nabla f=\lambda\nabla g+\mu\nabla h$ , giving five equations (three component equations plus the two constraints) in the five unknowns $x,y,z,\lambda,\mu$ . The non-parallel gradient hypothesis on $\nabla g$ and $\nabla h$ matters: if the two constraint surfaces are tangent at a point, the argument for why $\nabla f$ must lie in their span breaks down.

Examples

Extremizing $f(x,y,z)=x+2y+3z$ subject to $x^2+y^2+z^2=1$ and $x-y=0$ gives candidates at $(0.408,0.408,0.816)$ and its negation, the great-circle intersection of a sphere and a plane through the origin.