(*At a local minimum, the graph sits above its tangent plane*)

Show[ContourPlot3D[z==x^2+y^2,{x,-2,2},{y,-2,2},{z,-2,2},AxesOrigin->{0,0,0}],ParametricPlot3D[{x,y,0},{x,-2,2},{y,-2,2},PlotStyle->Orange]]

(*At a local maximum, the graph sits below its tangent plane*)

Show[ContourPlot3D[-z==x^2+y^2,{x,-2,2},{y,-2,2},{z,-2,2},AxesOrigin->{0,0,0}],ParametricPlot3D[{x,y,0},{x,-2,2},{y,-2,2},PlotStyle->Orange]]

(*Examples of Saddle Points with tangent planes shown*)

Show[ContourPlot3D[z==x^2-y^2,{x,-2,2},{y,-2,2},{z,-2,2},AxesOrigin->{0,0,0}],ParametricPlot3D[{x,y,0},{x,-2,2},{y,-2,2},PlotStyle->Orange]]

Show[ContourPlot3D[z==x^3+y^3,{x,-2,2},{y,-2,2},{z,-2,2},AxesOrigin->{0,0,0}],ParametricPlot3D[{x,y,0},{x,-2,2},{y,-2,2}]]

Show[ContourPlot3D[z==2*x*y -0.5(x^4+y^4)+1,{x,-2,2},{y,-2,2},{z,-1,3},AxesOrigin->{0,0,0}],ParametricPlot3D[{x,y,1},{x,-2,2},{y,-2,2}]]