There is the function $f(x,y,z)= z^{2}-yz-xz+z-2xy-2y+x^{2}+2x$ and the constraint function given by: $g(x,y,z)=3z-3y+z-1$. Find the conditional extremum of $f.$
The Lagrange function is given by:
$F(x,y,z, \lambda)=f(x,y,z)+\lambda \cdot g(x,y,z)$
So the necessary condition:
\begin{cases}g(x,y,z)=0 \\ \nabla f= \lambda \nabla g \end{cases}
The solution is given by $(x,y,z)=(1,1,1)$
How to check whether the point $(1,1,1)$ is a minimum, maximum or a stationary point? I mean how the sufficient conditions in this case are expressed.