$(1).$ Consider the following contraints:

$x_1+x_{2}9$

$x_{2}3$

$x_1-x_{2}0$

$x_{1}0$

$x_{2}0$

$($a$).$ Sketch the feasible region.

$($b$).$ Is there a MIN objective function that will have $x_1=2$ and $x_2=2$ as the unique optimal solution for

the constraints? If so provide one example. If not, give a brief explanation.

$($c$).$ Is there a MIN objective function that will have $x_1=3$ and $x_2=3$ as the unique optimal solution for

these contraints? If so provide one example. If not, give a brief explanation.

$($d$).$ For the objective function maxz$=2x_1+2x_2$ which of the $4$ cases do we have $($see page $67$ of textbook$)$:

$($i$)$The LP has a unique optimal solution

$($ii$)$ The LP has multiple $($alternative$)$ optimal solutions

$($iii$)$ The LP is infeasible

$($iv$)$ The LP is unbounded

$($e$).$ Is there a MAX objective function for whichan LP with this constraints is unbounded? Explain briefly.