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?

Hint: Look at the level sets $C_k=\{(x,y)\mid x+y=k\}$. Note that your objective function has value $k$ on $C_k$.
What's the largest $k$ such that $C_k$ meets $\overline{B(0,1)}$?
