Find the absolute extrema for f(x, y)=4-x^2-y^4+1/2y^2 over the closed disk D:x^2+y^2 is less than or equal to 1

Find the critical points of :

All three points lie within , and takes on values of

Now check for extrema on the boundary of . Convert to polar coordinates:

Find the critical points of :

where is any integer. There are some redundant critical points, so we'll just consider , which gives

which gives values of

So altogether, has an absolute maximum of 65/16 at the points (0, -1/2) and (0, 1/2), and an absolute minimum of 3 at (-1, 0).