 Failure of multiplier stationarity at a singular constraint

The nonzero gradient condition in the equality Lagrange multiplier rule cannot simply be dropped. Minimize $f(x,y)=x$ subject to $g(x,y)=x^2+y^2=0$. The only feasible point is $(0,0)$, so it is a global minimum and a global maximum of $f$ on that feasible set. Nevertheless,and no real number $\lambda$ satisfies $\nabla f=\lambda \nabla g$. The point is singular for this representation of the optimization constraint; the usual constraint qualification fails. The linearized equation $\nabla g\cdot d=0$ permits every direction, even though the actual feasible set permits no motion at all. This example explains why blindly solving the multiplier equations can miss an optimum.