I am struggling with the concept of boundedness in solving several optimization problems. When I go through the theory I understand what it means. However when I look at an example I am struggling to figure out where to begin from. For example in the following optimisation problem there are two constraints and I am trying to maximize the objective function.
$\max(x,y,z) x + 2y − 3z$;
such that $x^2 − 3xy + z^2 \le 5$;
$4x^2 − 2xy + y^2 + 4xz + 10z^2 = 100$
To figure out if the constraints are bounded I tried to solve the equations and came up with the following bounds
$0 < x \le 6,$
$0 < y \le 10,$
$-\sqrt{5} \le z \le 0.$
I have gotten a feeling these are not right as I think my understanding of how to find the bounds are not clear. It would be much appreciated if I can get some ideas or information on bounds including any resource materials that might help me to better understand the concept.
Thanks.