 Solution

For minimization, the optimization Lagrangian isIts coefficient of $z$ is $-1-\lambda <0$, so its infimum over unrestricted $z$ is $-\infty$ for every allowable Lagrange multiplier. There is no finite global minimizer of the optimization Lagrangian to which the Lagrangian sufficiency theorem could apply.

The primal minimization problem is also unbounded: set $x=0$, $y=\sqrt{5}$ and $z=t\ge 0$. These points are feasible, and $3y-z=3\sqrt{5}-t\to -\infty$. Hence