Question

Find the absolute maximum and absolute minimum of the function f(x,y)=2x^3+y^4 on the region {(x,y)|x^2+y^2≤25}.

Answer 1

for any constant x, f(x,y) is just k + y^4, so it will have a local minimum at y=0

for any constant y, f(x,y) is just k+2x^3, which has no relative extrema
clearly Fx = Fy = 0 at (0,0) but that is not an extremum.
So, again, check the values on the boundary.
It should not take too long to satisfy yourself that the minimum is
f(-5,0) = -250
f(0,±5) = 625