1. Consider the function $f(x, y) = ax^2y + bxy + 2xy^2 + c$.
(a) Determine values of the constants $a$, $b$, and $c$ such that $f$ 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 the set $S = \{(x, y): x \ge 0, y \ge 0, 2x + y \le 4\}$.