It sounds like you are doing the math work in the probelm right, which is good. One way to check your critical points is using the second partials of the function, which can be a pain, but in this case isnt too bad. Just remember to substitute a value for in terms of and in the equation before you try to take a derivative of it.
-If Fxx and D(x,y,z) are positve then its a min.
-If Fxx is negative and D(x,y,z) is positive then the point is a max.
-If D(x,y,z) is less than 0 then the point is a saddle point.
*note: where means take the partial derivative of F with respect to x twice.