Just examine these four cases: x=-1, x=1, y=-1, y=1.

For example,

x=-1

f(-1,y)=1-4y^2 + y^4

df(-1,y)/dy = 4y^3 -8y

Equate this to 0 and solve for y.

4y(y^2 - 2)=0

y=0, y=sqrt(2) [Ignore, since it is outside the region]

Do these for all cases, and also check the endpoints (-1,1), (-1,-1), (1,1), (1,-1)

Of all these values, find out which one is the highest and the lowest.