Any $(x, y, z)$ can satisfy the $5x^2+2y^2+6z^2-6xy-2xz+2yz<0$? Please tell me whether there any $(x, y, z)$ 
which can satisfy the $5x^2+2y^2+6z^2-6xy-2xz+2yz<0$ ?
No process or just solve it by calculator are both fine.
Thank you.

Consider that you search for the extremum of $$F=5x^2+2y^2+6z^2-6xy-2xz+2yz$$
$$\frac{\partial F}{\partial x}=10 x-6 y-2 z=0 \qquad \frac{\partial F}{\partial y}=-6 x+4 y+2 z=0 \qquad \frac{\partial F}{\partial z}=-2 x+2 y+12 z=0$$
The only solution is $x=y=z=0$ so the minimum value of $F$ is $0$. 
Then $F$ is non-negative for any $(x,y,z)$.
