Using Lagrange Multipliers to find max/min of this function relative to a condition set (inequality)

Question

I was wondering how should I treat the inequality conditions for this exercise; to be more precise, how to form them. Is deducing from $$ x \in [0,2] $$ and $$ y\in[0,4] $$ that $$ x+y \le 6 $$ alright? Can I use this as my condition? If so, if I begin the search on two separate case:

i) (x,y) belongs to the interior of S (x+y<6) and

ii) (x,y) belongs to boundary of S (x+y=6) correct?

For i) I find the point for which $\nabla f = 0$ then check if it provides a min/max and for ii) I apply Lagrange multipliers theorem to find the points?