Answer
(a) True
(b) False
(c) False
(d) False
Work Step by Step
(a) If $D$ is closed and bounded, then by Theorem 3, $f$ takes on both a minimum and a maximum value on $D$. Though the statement only mentions the maximum, it is true.
(b) If $D$ is neither closed nor bounded, then we can consider an open disk enclosing a point $P$ in the domain. By the definition of Local Extreme Values (on page 811) $f$ can have local extremum. Thus, the statement is false.
(c) The domain $D$ defined by $0 \le x \le 1$, $0 \le y \le 1$ is closed and bounded. By Theorem 3, $f$ takes on both a minimum and a maximum value on $D$. Thus, the statement is false.
(d) The domain $D$ is defined on the open quadrant:
$\left\{ {\left( {x,y} \right):x > 0,y > 0} \right\}$
Then we can consider an open disk enclosing a point $P$ in the domain. By the definition of Local Extreme Values (on page 811) $f$ can take on local extremum. Thus, the statement is false.
