Find the absolute maximum and minimum values of the function f(x, y) = x2 + xy + y2 on the disc x2 + y2 ≤ 1

Critical points occur when both partial derivatives vanish; this happens when



The function has Hessian



which at has . And since , it follows that a local minimum occurs at with a value of .

Meanwhile, we can parameterize the boundary by



with . So



which has critical points when :



We only have 4 points to worry about:

Now,






So we find that an absolute minimum occurs at with a value of , and two more extrema occur along the boundary when and , i.e at the points and , both with the same maximum value of .