$.$ Consider a linear program in variables x$1$ and x$2$ with the constraints:

$$x$1+$ x$2>=1$

x$1>=0$

$($a$)$ Plot the feasible region of the linear program.

$($b$)$ Consider minimizing the linear function $\backslash$alpha x$1+$ x$2$ over this feasible region. For each of

the cases below, ffnd all values of $\backslash$alpha such that:

i$.$ the optimal solution is unique

ii$.$ there are multiple optimal solutions with ffnite optimal objective value

iii. the optimal objective value is unbounded