(*Polytope 3*)
f[x_,y_]:=1+1/(x*y)+y+x
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
CommenRoots[[1]]&&f->(f[x,y]/.CommenRoots[[1]])
CommenRoots[[2]]&&f->(f[x,y]/.CommenRoots[[2]])
CommenRoots[[3]]&&f->(f[x,y]/.CommenRoots[[3]])

(*Polytope 4a*)
f[x_,y_]:=1+1/y+y+1/x+x
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
CommenRoots[[1]]&&f->(f[x,y]/.CommenRoots[[1]])
CommenRoots[[2]]&&f->(f[x,y]/.CommenRoots[[2]])
CommenRoots[[3]]&&f->(f[x,y]/.CommenRoots[[3]])
CommenRoots[[4]]&&f->(f[x,y]/.CommenRoots[[4]])

(*Polytope 4b*)
f[x_,y_]:=1+y+1/x*y+x+1/y
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
N[CommenRoots[[1]]]&&f->(N[f[x,y]/.CommenRoots[[1]]])
N[CommenRoots[[2]]]&&f->(N[f[x,y]/.CommenRoots[[2]]])
N[CommenRoots[[3]]]&&f->(N[f[x,y]/.CommenRoots[[3]]])
N[CommenRoots[[4]]]&&f->(N[f[x,y]/.CommenRoots[[4]]])

(*Polytope 4c*)
f[x_,y_]:=1+1/x*y+y+x*y+1/y
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
CommenRoots[[1]]&&f->(f[x,y]/.CommenRoots[[1]])
CommenRoots[[2]]&&f->(f[x,y]/.CommenRoots[[2]])
CommenRoots[[3]]&&f->(f[x,y]/.CommenRoots[[3]])
CommenRoots[[4]]&&f->(f[x,y]/.CommenRoots[[4]])

(*Polytope 5a*)
f[x_,y_]:=1+(1/x)*y+y+1/x+x+1/y
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
CommenRoots[[1]]&&f->(FullSimplify[f[x,y]/.CommenRoots[[1]]])
CommenRoots[[2]]&&f->(FullSimplify[f[x,y]/.CommenRoots[[2]]])
N[CommenRoots[[3]]]&&f->(N[f[x,y]/.CommenRoots[[3]]])
N[CommenRoots[[4]]]&&f->(N[f[x,y]/.CommenRoots[[4]]])
N[CommenRoots[[5]]]&&f->(N[f[x,y]/.CommenRoots[[5]]])

(*Polytope 5b*)
f[x_,y_]:=1+1/x*y+y+x*y+1/x+1/y
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
N[CommenRoots[[1]]]&&f->(N[f[x,y]/.CommenRoots[[1]]])
N[CommenRoots[[2]]]&&f->(N[f[x,y]/.CommenRoots[[2]]])
N[CommenRoots[[3]]]&&f->(N[f[x,y]/.CommenRoots[[3]]])
N[CommenRoots[[4]]]&&f->(N[f[x,y]/.CommenRoots[[4]]])
N[CommenRoots[[5]]]&&f->(N[f[x,y]/.CommenRoots[[5]]])

(*Polytope 6a*)
f[x_,y_]:=1/y+x*1/y+1/x+1+x+1/x*y+y
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
CommenRoots[[1]]&&f->(f[x,y]/.CommenRoots[[1]])
CommenRoots[[2]]&&f->(f[x,y]/.CommenRoots[[2]])
CommenRoots[[3]]&&f->(FullSimplify[f[x,y]/.CommenRoots[[3]]])
CommenRoots[[4]]&&f->(FullSimplify[f[x,y]/.CommenRoots[[4]]])
CommenRoots[[5]]&&f->(f[x,y]/.CommenRoots[[5]])
CommenRoots[[6]]&&f->(f[x,y]/.CommenRoots[[6]])

(*Polytope 6b*)
f[x_,y_]:=1+1/x*y+y+x*y+1/x+x+1/y
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
CommenRoots[[1]]&&f->(f[x,y]/.CommenRoots[[1]])
CommenRoots[[2]]&&f->(f[x,y]/.CommenRoots[[2]])
CommenRoots[[3]]&&f->(f[x,y]/.CommenRoots[[3]])
CommenRoots[[4]]&&f->(f[x,y]/.CommenRoots[[4]])
CommenRoots[[5]]&&f->(f[x,y]/.CommenRoots[[5]])
CommenRoots[[6]]&&f->(f[x,y]/.CommenRoots[[6]])

(*Polytope 6c*)
f[x_,y_]:=1+1/x*y+y+x*y+1/x+1/(x*y)+1/y
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
N[CommenRoots[[1]]]&&f->(N[f[x,y]/.CommenRoots[[1]]])
N[CommenRoots[[2]]]&&f->(N[f[x,y]/.CommenRoots[[2]]])
N[CommenRoots[[3]]]&&f->(N[f[x,y]/.CommenRoots[[3]]])
N[CommenRoots[[4]]]&&f->(N[f[x,y]/.CommenRoots[[4]]])
N[CommenRoots[[5]]]&&f->(N[f[x,y]/.CommenRoots[[5]]])
N[CommenRoots[[6]]]&&f->(N[f[x,y]/.CommenRoots[[6]]])

(*Polytope 6d*)
f[x_,y_]:=1/(x*y^2)+1/(x*y)+1/x+1+1/x*y+y+x*y
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
CommenRoots[[1]]&&f->(f[x,y]/.CommenRoots[[1]])
N[CommenRoots[[2]]]&&f->(N[f[x,y]/.CommenRoots[[2]]])
N[CommenRoots[[3]]]&&f->(N[f[x,y]/.CommenRoots[[3]]])
N[CommenRoots[[4]]]&&f->(N[f[x,y]/.CommenRoots[[4]]])
N[CommenRoots[[5]]]&&f->(N[f[x,y]/.CommenRoots[[5]]])
N[CommenRoots[[6]]]&&f->(N[f[x,y]/.CommenRoots[[6]]])

(*Polytope 7a*)
f[x_,y_]:=1+1/x*y+y+x*y+1/x+x+1/(x*y)+1/y
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
N[CommenRoots[[1]]]&&f->N[f[x,y]/.CommenRoots[[1]]]
N[CommenRoots[[2]]]&&f->N[f[x,y]/.CommenRoots[[2]]]
N[CommenRoots[[3]]]&&f->N[f[x,y]/.CommenRoots[[3]]]
N[CommenRoots[[4]]]&&f->N[f[x,y]/.CommenRoots[[4]]]
N[CommenRoots[[5]]]&&f->N[f[x,y]/.CommenRoots[[5]]]
N[CommenRoots[[6]]]&&f->N[f[x,y]/.CommenRoots[[6]]]
N[CommenRoots[[7]]]&&f->N[f[x,y]/.CommenRoots[[7]]]

(*Polytope 7b*)
f[x_,y_]:=1+1/x*y+y+x*y+1/x+1/(x*y)+1/y+1/(x*y^2)
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
N[CommenRoots[[1]]]&&f->N[f[x,y]/.CommenRoots[[1]]]
N[CommenRoots[[2]]]&&f->N[f[x,y]/.CommenRoots[[2]]]
N[CommenRoots[[3]]]&&f->N[f[x,y]/.CommenRoots[[3]]]
N[CommenRoots[[4]]]&&f->N[f[x,y]/.CommenRoots[[4]]]
N[CommenRoots[[5]]]&&f->N[f[x,y]/.CommenRoots[[5]]]
N[CommenRoots[[6]]]&&f->N[f[x,y]/.CommenRoots[[6]]]
N[CommenRoots[[7]]]&&f->N[f[x,y]/.CommenRoots[[7]]]

(*Polytope 8a*)
f[x_,y_]:=1+1/x*y+y+x*y+1/x+x+1/(x*y)+1/y+x*1/y
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
CommenRoots[[1]]&&f->(FullSimplify[f[x,y]/.CommenRoots[[1]]])
CommenRoots[[2]]&&f->(FullSimplify[f[x,y]/.CommenRoots[[2]]])
CommenRoots[[3]]&&f->(FullSimplify[f[x,y]/.CommenRoots[[3]]])
CommenRoots[[4]]&&f->(FullSimplify[f[x,y]/.CommenRoots[[4]]])
CommenRoots[[5]]&&f->(f[x,y]/.CommenRoots[[5]])
CommenRoots[[6]]&&f->(f[x,y]/.CommenRoots[[6]])
CommenRoots[[7]]&&f->(f[x,y]/.CommenRoots[[7]])
CommenRoots[[8]]&&f->(f[x,y]/.CommenRoots[[8]])

(*Polytope 8b*)
f[x_,y_]:=1+1/x*y+y+x*y+1/x+x+1/(x*y)+1/y+1/(x*y^2)
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
CommenRoots[[1]]&&f->(f[x,y]/.CommenRoots[[1]])
CommenRoots[[2]]&&f->(f[x,y]/.CommenRoots[[2]])
CommenRoots[[3]]&&f->(f[x,y]/.CommenRoots[[3]])
CommenRoots[[4]]&&f->(f[x,y]/.CommenRoots[[4]])
N[CommenRoots[[5]]]&&f->N[f[x,y]/.CommenRoots[[5]]] (*Numerical Value*)
N[CommenRoots[[6]]]&&f->N[f[x,y]/.CommenRoots[[6]]] (*Numerical Value*)
N[CommenRoots[[7]]]&&f->N[f[x,y]/.CommenRoots[[7]]] (*Numerical Value*)
N[CommenRoots[[8]]]&&f->N[f[x,y]/.CommenRoots[[8]]] (*Numerical Value*)

(*Polytope 8c*)
f[x_,y_]:=1+1/x*y+y+x*y+1/x+1/(x*y)+1/y+1/(x*y^2)+1/(x*y^3)
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
CommenRoots[[1]]&&f->(f[x,y]/.CommenRoots[[1]])
CommenRoots[[2]]&&f->(f[x,y]/.CommenRoots[[2]])
CommenRoots[[3]]&&f->(f[x,y]/.CommenRoots[[3]])
CommenRoots[[4]]&&f->(f[x,y]/.CommenRoots[[4]])
N[CommenRoots[[5]]]&&f->N[f[x,y]/.CommenRoots[[5]]] (*Numerical Value*)
N[CommenRoots[[6]]]&&f->N[f[x,y]/.CommenRoots[[6]]] (*Numerical Value*)
N[CommenRoots[[7]]]&&f->N[f[x,y]/.CommenRoots[[7]]] (*Numerical Value*)
N[CommenRoots[[8]]]&&f->N[f[x,y]/.CommenRoots[[8]]] (*Numerical Value*)

(*Polytope 9*)
f[x_,y_]:=1+1/x*y+y+x*y+x^2*y+1/x+x+1/(x*y)+1/y+1/(x*y^2)
CommenRoots:=Solve[D[f[x,y],x]==0&&D[f[x,y],y]==0,{x,y}]
CommenRoots[[1]]&&f->(f[x,y]/.CommenRoots[[1]])
CommenRoots[[2]]&&f->(f[x,y]/.CommenRoots[[2]])
CommenRoots[[3]]&&f->(f[x,y]/.CommenRoots[[3]])
CommenRoots[[4]]&&f->(f[x,y]/.CommenRoots[[4]])
CommenRoots[[5]]&&f->(f[x,y]/.CommenRoots[[5]])
CommenRoots[[6]]&&f->(f[x,y]/.CommenRoots[[6]])
FullSimplify[CommenRoots[[7]]]&&f->FullSimplify[f[x,y]/.CommenRoots[[7]]]
FullSimplify[CommenRoots[[8]]]&&f->FullSimplify[f[x,y]/.CommenRoots[[8]]]
FullSimplify[CommenRoots[[9]]]&&f->FullSimplify[f[x,y]/.CommenRoots[[9]]]