Is there an absolute maximum value of this function?

[pantin]

Homework Statement

Consider the function f:R^2->R defined by f(x,y)=[e^(x+y)]-y+x. Is there an absolute maximum value of f on the set s={(x,y):/x/+/y/<=2}? Justify.

note, /x/ is the absolute value of x.

Homework Equations

a. If f is con't, it takes compact sets to compact sets.

b.Extreme value thm: Suppose s belongs to R^n is compact and f: s->R is continuous. then f has an absolute min value and an absolute max value on S

The Attempt at a Solution

My idea is
=>show s is bounded and closed => therefore compact => f con't maps compact sets to compact sets => EVT

but I encounted problem when finding boundary of s
/x/+/y/<=2 that means /x/<=2 when y=0 and /y/<=2 when x=0. then can I find the boundary here by constructing a circle with x and y? because I saw other example did it on this way.
Help..

 

[morphism]

Relevant thread: https://www.physicsforums.com/showthread.php?t=261186.