Answer
a) See the explanation below.
b) See the explanation below.
c) See the explanation below.
d) See the explanation below.
e) See the explanation below.
Work Step by Step
a) This means that $f(x,y) \leq f(a,b)$ when $(x,y)$ is near $(a,b)$ and $f$ has a local maximum at $(a,b)$.
b) This means that $f(x,y) \leq f(a,b)$ when $(x,y)$ for all points $(x,y)$ a set of real numbers in the domain $D$ and $f$ has an absolute maximum at $(a,b)$.
c) This means that $f(x,y) \geq f(a,b)$ when $(x,y)$ is near $(a,b)$ and $f$ has a local minimum at $(a,b)$.
d) This means that $f(x,y) \geq f(a,b)$ when for all points $(x,y)$, a set of real numbers in the domain $D$ and $f$ has an absolute minimum at $(a,b)$.
e) This means that $f(x,y)$ has neither a local maximum nor a local minimum or both are in opposite directions when $(x,y)$ is near $(a,b)$.
