Determine values of the constants a b and c such that f x y

Question

Question: (a) Determine values of the constants a, b, and c such that f (x, y) = ax²y + bxy + 2xy² + c has a local minimum at the point (2/3, 1/3) with local minimum value -1/9.

(b) With the values of a, b, and c found in part (a), find the maximum and minimum values of f over S = {(x, y) : x ≥ 0, y ≥ 0, 2x + y ≤ 4}.