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 condtion:
\begin{cases}g(x,y,z)=0 \\ \nabla f= \lambda \nabla g \end{cases}
The solition 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.
Best Answer
the type of point is given by the Hessian matrix definiteness. From this article, (https://en.wikipedia.org/wiki/Hessian_matrix) the rules are: "If the Hessian is positive definite at x, then f attains an isolated local minimum at x. If the Hessian is negative definite at x, then f attains an isolated local maximum at x. If the Hessian has both positive and negative eigenvalues then x is a saddle point for f. Otherwise the test is inconclusive. This implies that at a local minimum the Hessian is positive-semi-definite, and at a local maximum the Hessian is negative semi-definite."