Show[
ContourPlot3D[3x^2+2y^2+z^2==9,{x,-3,10},{y,-3,10},{z,-3,10},ContourStyle->Orange,Mesh->False],ContourPlot3D[x^2+y^2+z^2-8x-6y-8z+24==0,{x,-3,10},{y,-3,10},{z,-3,10},ContourStyle->Blue,Mesh->False]
]

Manipulate[Show[
ContourPlot3D[x^2-y^2+z^2==6,{x,-6,6},{y,-6,6},{z,-6,6},ContourStyle->{Orange,Opacity[h]},Mesh->False,AxesOrigin->{0,0,0}],ContourPlot3D[x^2+y^2+z^2==14,{x,-6,6},{y,-6,6},{z,-6,6},ContourStyle->{Blue,Opacity[s]},Mesh->False],ParametricPlot3D[{(Sqrt[10])*Cos[t],2,(Sqrt[10])*Sin[t]},{t,0,2*π},PlotStyle->{Red,Thick,Opacity[int]}],ParametricPlot3D[{(Sqrt[10])*Cos[t],-2,(Sqrt[10])*Sin[t]},{t,0,2*π},PlotStyle->{Red,Thick,Opacity[int]}]
],{{h,1},0,1},{{s,1},0,1},{{int,0},0,1}]

(*These are examples of what can happen when the determinant of the Hessian is zero. The first case shows that it can still have a local maximum (removing the minus outside of the parenthesis makes it a minimum). The second case shows that it could be a saddle point.*)

Show[ParametricPlot3D[{x,y,-(x-y)^2},{x,-.5,.5},{y,-.5,.5}],ParametricPlot3D[{0,0,t},{t,-1,1}]]

Show[ParametricPlot3D[{x,y,x^3+y^3},{x,-.5,.5},{y,-.5,.5},BoxRatios->{1.1,1.1,1}],ParametricPlot3D[{0,0,t},{t,-1,10}]]