Find the absolute maximum and minimum values of f on the set d. f(x, y) = 9 + xy − x − 2y, d is the closed triangular region with vertices (1, 0), (5, 0), and (1, 4)

which tells you that the only critical point of occurs at (2, 1), which does lie within the region . At this point, we get .

Next we check along the boundaries of . They are the lines with , with , and with .

If , then , which is monotonically decreasing and must therefore attain its maximum at and minimum at . We get and .

If , then , which is also monotonically decreasing and attains its maximum at and minimum at . We get and .

If , then . We have , which suggests an extremum occurs at (3, 2). We get .

So has a minimum value of 4 at (1, 4) and (5, 0), and a maximum value of 8 at (1, 0) and (3, 2).

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