I'm trying to find the local extrema of a multi-variable function...kinda lost as to whether or not my approach is correct. If someone can look this over and tell me if i'm wrong, and if so, whats the right approach $$ F(x,y,z) = (x+y)^2 + (x+z)^2 + (y+z)^2 $$ My current approach is using partial derivatives to form a system of equations...
$\frac{\partial}{\partial x} = 4x+2y+2z$
$\frac{\partial}{\partial y} = 2x+4y+2z$
$\frac{\partial}{\partial z} = 2x+2y+4z$
solving for this system gives $F(x,y,z)=(0,0,0)$
Then using the gradient of the function to make a Hessian Matrix gives
$\nabla f(x)=[4x+2y+2z,\ 2x+4y+2z,\ 2x+2y+4z]$
Hessian Matrix
$$ H(x)= \begin{matrix} 4 & 2 & 2 \newline 2 & 4 & 2 \newline 2 & 2 & 4 \newline \end{matrix} $$
Finding the determinants of all principal minors...
$det(H_1)=4 ,$ $det(H_2)=12 ,$ $det(H_3)=32 $
since all determinants are positive $H(0,0,0)$ is positive definite
and $(x,y,z)=(0,0,0)$ is a local minima
Appreciate the help!