We want to find the absolute maximum value of f(x, y) = x 3 − 3x − y 2 + 12 on the domain d = {(x, y) | x 2 + y 2 ≤ 9, y ≥ 0}, a half disk. (a) (3 pts) first, find the critical points of f, and determine if any of the critical points is a local maximum in the domain d. (b) (3 pts) then, find the (absolute) maximum value of f along the boundary of d. the bottom edge of d and along the top arc of d. (note: some of the values might be a bit messy. give solutions to two decimal places.) (c) (2 pts) using parts (a) and (b), find the absolute maximum value of f on d

Critical points occur where both first order derivatives vanish:




So we have two critical points, (-1, 0) and (1, 0), both of which lie in the domain (on the boundary ). At these points, we get values of and .

If we focus on just the boundaries:

(1) If , then , and , so there are no other critical points along this boundary.

(2) If we take and , we can parameterize the semicircular arc by considering . So we can write





In the first case, we have either or . In the second, we can solve the quadratic to find or . At each critical point, we get






Over the domain , we get an absolute maximum value of 30 at , which corresponds to the point .