showing function is non-negative

showing function is non-negative

ok, since it's been so long when I took Calculus, I just wanna make sure
I'm not doing anything wrong here.
Given $f:\mathbb{R}^2\rightarrow \mathbb{R}$ defined as $f(x,y)=x+y-ye^x$.
I would like to show that the function is nonnegative in the region
$x+y\leq 1, \;\;x\geq 0, \;\;y\geq 0$.
Now my game plan is as follows: 1. Show the function is non-negative on
the boundary of the region 2. Show the function takes a positive value in
the interior of the region 3. Show that the function has no critical
points in the interior of the region 4. By continuity the function is
non-negative everywhere in the region.
Is the above sufficient or am I doing something wrong? Would there be a
better way to show this?