I want to find the maximum of $f(x,y) = x^ae^{-x}y^be^{-y}$ on the triangle given by $x\geq0$, $y\geq0$, and $x+y\leq1$ in terms of $a$ and $b$ such that $a,b>0$.
I can see that the vertices of the triangle are $(0,0)$, $(1,0)$, and $(0,1)$. To check whether there is a maximum in the interior of the triangle, would I just check to find points where both partials vanish, or do I need to apply Lagrange multipliers? If so, what would I use for the constraint function? On the boundaries, I know to check the values at the three vertices, but what about all the other points on the boundaries?
On two of the three boundaries, the function is just zero. On the last one, you know $x + y = 1$, so you can rewrite your function as a one-variable function
$$f(x, y) = f(x, 1 - x) = x^a e^{-x} (1 - x)^b e^{-(1 - x)} = \frac 1 e x^a (1 - x)^b$$
which isn't too hard to maximize.
Then for the interior, try to identify the critical points where $f_x = f_y = 0$.