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