Let $f(x,y,z)=xyz+z$ be the objective function, and suppose that $0\leq6-x^2-y^2-z$, $0\leq x$, $0\leq y$, $0\leq z$ are the four constraint functions. I must determine if the constraint $0\leq 6-x^2-y^2-z$ is binding for any solution to this maximization problem, i.e. if there is a solution $w_0=(x_0,y_0,z_0)$ for which $6=x_0^2+y_0^2+z_0$.
Because the objective function and the constraint functions are both weakly concave and differentiable, and since the interior of the constraint set is non-empty, Kuhn-Tucker tells us that $w_0$ solves the maximization problem iff $w_0$ satisfies the first-order conditions (FOCs) of the Lagrangian.
Hence, to see if this particular constraint is binding, I feel like I need to first find the set of points that solve the Lagrangian FOCs. However, solving this system of equations is nearly impossible by hand, and so I'm wondering if there is another method to check if this constraint is binding for some solution.