Sanity check about a linear programming problem given the linear program:

minimize $x+y$

subject to,

$ax+by \leq 1$

$x,y \geq 0$

I need to find real numbers $a,b \in \mathbb{R}$ such that the program (a) is infeasible, (b) is unbounded, and (c) has a unique optimal solution.

Since we are asked to minimize the sum of two non-negative numbers the solution for all $a,b \in \mathbb{R}$ is $(x,y)=(0,0)$, right?

I cannot see any potential values of $a,b$ which would make the program infeasible or unbounded, because we are always looking to minimize the sum of non-negative numbers.
I’d say the program is always feasible because x = y = 0 is a solution.

Most pairs a, b give a unique solution. Only certain pairs don’t. 

Now it depends on the definition of “unbounded”. If we talk about the goal function, the minimum is always bounded by 0. If we talk about the points (x, y), they are unbounded if a &lt;= 0 or b &lt;= 0.