Detrmine the maximum value and the min value of f(x,y)= x^2+y^2-x-y on the closed unit disc D: x^2+y^2<=1.
If a max or min occurs in the interior of the disk, then they will occur where f_x= 2x- x= 0 and f_y= 2y- y= 0.
But it is also possible that the max and/or min occurs on the bounding circle. On that circle, x= cos(t), y= sin(t) so that f(x,y)= f(t)= 1- cos(t)- sin(t). The max or min will occur where f'(t)= sin(t)- cos(t)= 0