g[x_,y_]=(3/4) y^2+(1/24) y^3-(1/32)y^4-x^2;

D[g[x,y],x]

D[g[x,y],y]

Factor[%]

gradg[x_,y_]=Grad[g[x,y],{x,y}]

Solve[Evaluate[Grad[g[x,y],{x,y}]=={0,0},{x,y}]]

{p[1],p[2],p[3]}={{0,-3,g[0,-3]},{0,0,g[0,0]},{0,4,g[0,4]}}

ph[j_]:=Show[Graphics3D[{PointSize[Large],Blue,Point[p[j]]}],Plot3D[p[j][[3]],{x,-1+p[j][[1]],1+p[j][[1]]},{y,-1+p[j][[2]],1+p[j][[2]]},Mesh->False,PlotStyle->{Opacity[0.4],LightGray},BoundaryStyle->{Thick,Blue}]];

Show[Plot3D[(3/4) y^2+(1/24) y^3-(1/32)y^4-x^2,{x,-7,7},{y,-7,7},RegionFunction->Function[{x,y,z},(3/4) y^2+(1/24) y^3-(1/32)y^4-x^2≥ -8]],ph[1],ph[2],ph[3]]

Show[Plot3D[(3/4) y^2+(1/24) y^3-(1/32)y^4-x^2,{x,-7,7},{y,-7,7},RegionFunction->Function[{x,y,z},(3/4) y^2+(1/24) y^3-(1/32)y^4-x^2≥ -8],PlotStyle->Opacity[0.3],Mesh->False],ParametricPlot3D[{{t,0,g[t,0]},{0,t,g[0,t]}},{t,-2,2},Mesh->False,PlotStyle->{{Blue,Thick},{Blue,Thick}}]]

Show[ContourPlot[(3/4) y^2+(1/24) y^3-(1/32)y^4-x^2,{x,-7,7},{y,-7,7},ContourShading->False,ContourLabels->True,Contours->Table[-10+k,{k,1,16}]],Graphics[{PointSize[Large],Blue,Thick,Point[{{0,-3},{0,0},{0,4}}]}]]

Show[%,StreamPlot[gradg[x,y],{x,-7,7},{y,-7,7}]]

Plot3D[(3/4) y^2+(1/24) y^3-(1/32)y^4-x^2,{x,-7,7},{y,-7,7},RegionFunction->Function[{x,y,z},(3/4) y^2+(1/24) y^3-(1/32)y^4-x^2≥ -8],MeshFunctions->(#3 &),MeshStyle->{Blue,Thick},PlotPoints->20]

Show[ContourPlot3D[{z== -15,z== g[x,y]},{x,-7,7},{y,-7,7},{z,-15,10},RegionFunction->Function[{x,y,z},(3/4) y^2+(1/24) y^3-(1/32)y^4-x^2≥ -8],MeshFunctions->(g[#1,#2] &),ContourStyle->{{Opacity[0.4],LightGreen},{Opacity[0.4],LightBlue}},PlotPoints->20,AspectRatio->Automatic,PlotRange->All],ContourPlot3D[{z== -15,z== g[x,y]},{x,-7,7},{y,-7,7},{z,-15,10},RegionFunction->Function[{x,y,z},(3/4) y^2+(1/24) y^3-(1/32)y^4-x^2≥ -8],MeshFunctions->(g[#1,#2] &),Mesh->{{0}},MeshStyle->{Thick,Black,Dashed},ContourStyle->{{Opacity[0.4],LightGreen},{Opacity[0.4],LightBlue}},PlotPoints->20,AspectRatio->Automatic,PlotRange->All]]