Maximize the function $x+y$ on the closure of the unit ball 
Find the maximum of $\{x +y : (x,y) \in closure [B(0,1)] \}$. Here $B(0,1)=\{(x,y) \in \mathbb{R}^2 : x^2+y^2 \leq 1\}$

I can't proceed in this as to how to maximize this value. Will it lie on the boundary of the unit ball or somewhere inside the ball?

Consider this:
The point (0,0) lies in your ball. So your set contains 0 + 0 = 0.
It also contains 0 + 1 = 1 since (0,1) lies on the closure of B(0,1).
So what other values does your set have? And where is the biggest? 
