Through[{Minimize, Maximize}[{2 x^3 + y^4, Norm[{x, y}] <= 1}, {x, y}]]
to visualize
Plot3D[2 x^3 + y^4, {x, -1.4, 1.4}, {y, -1.4, 1.4},
RegionFunction -> ({x, y} \[Function] x^2 + y^2 < 1),
MeshFunctions -> {#3 &}]
For Question 2
RegionNearest[ImplicitRegion[x^2 + y^2 == z, {x, y, z}], {5, 5, 0}]