Plot3D[x^4+y^4-4x y+1, {x, -2, 2}, {y,-2,2}]

Show[
Plot3D[x^4+y^4-4x y+1, {x, -2, 2}, {y,-2,2}],ListPointPlot3D[{{-1,-1,1+1-4+1},{1,1,1+1-4+1},{0,0,1}}, Filling->Bottom, PlotStyle->PointSize[Large], ColorFunction->Function[{x,y,z},Hue[z/3]]]
]

Show[
Plot3D[x^4+y^4-4x y+1, {x, 0, 2}, {y,0,2}],ListPointPlot3D[{{1,1,1+1-4+1}}, Filling->Bottom, PlotStyle->PointSize[Large], ColorFunction->Function[{x,y,z},Hue[z/3]]],
ParametricPlot3D[{{1, 1+t, 1+(1+t)^4-4(1+t)+1},{1+t, 1,1+(1+t)^4-4(1+t)+1 }},{t, -1, 1}]
]

Show[
Plot3D[x^4+y^4-4x y+1, {x, .5, 1.5}, {y,.5,1.5}],ListPointPlot3D[{{1,1,1+1-4+1}}, Filling->Bottom, PlotStyle->PointSize[Large], ColorFunction->Function[{x,y,z},Hue[z/3]]],
ParametricPlot3D[{{1, 1+t, 1+(1+t)^4-4(1+t)+1},{1+t, 1,1+(1+t)^4-4(1+t)+1 }},{t, -1, 1}, ColorFunction->Function[{x,y,z}, .5]]
]

Show[
Plot3D[x^4+y^4-4x y+1, {x, -1, 1}, {y,-1,1}],ListPointPlot3D[{{0,0,1}}, Filling->Bottom, PlotStyle->PointSize[Large], ColorFunction->Function[{x,y,z},Hue[z/3]]],
ParametricPlot3D[{{0, t, (t)^4+1},{t, 0,(t)^4+1 }},{t, -1, 1}, ColorFunction->Function[{x,y,z}, .5]]
]

Show[
Plot3D[x^4+y^4-4x y+1, {x, -1, 1}, {y,-1,1}],ListPointPlot3D[{{0,0,1}}, Filling->Bottom, PlotStyle->PointSize[Large], ColorFunction->Function[{x,y,z},Hue[z/3]]],
ParametricPlot3D[{{0, t, (t)^4+1},{t, 0,(t)^4+1 }},{t, -1, 1}, ColorFunction->Function[{x,y,z}, .5]],
ParametricPlot3D[{t, t, t^4+(t)^4-4t^2+1},{t, -1, 1}, ColorFunction->Function[{x,y,z}, .1]]
]