You have the system:
(1) The critical points are called equilibrium points or fixed points. That's where all the derivatives are zero. You can plug in into each and see that this makes them all zero.
(2) You need to just start splitting out partials for the Jacobian. You know what the Jacobian is right? It's
Now is just so that right? How about ? Can you calculate the partials with respect to each x of ?. Do the same with the other two and you'll get the full Jacobian matrix. Once you get it, you need to evaluate it at the fixed point , that is . You can do that, and you get the linearized matrix . Now calculate all the eigenvalues of that matrix and then determine if the critical point at is stable or not depending on the signs of the eigenvalues.
Oh yea, all the secrets of the Universe can be found in differential equations.