Question $2.(15$ points$)$: Consider the following LP in standard form, with a single equality constraint:

$ma{x}_x{\displaystyle \underset{i=1}{\overset{n}{}}}{c}_i{x}_i$

$s.t.{\displaystyle \underset{i=1}{\overset{n}{}}}{a}_i{x}_i=b$

$x_{i}0,i=1,$dots,$n$

Suppose $b>0$ and $a_i>0$ for each $i=1,$dots,$n.$

A$)$ Write down all of the basic feasible solutions of $(2).$

B$)$ For each $i=1,$dots,$n,$ find a number $M_i,$ which may depend on $b$ and $a_i,$ such that any feasible solution

$x$ satisfies $x_i{M}_i.$

C$)$ If $c_{i}0$ for all $i=1,$dots,$n,$ then does the LP in $(2)$ have an optimal solution? If yes, what is it and what

is the optimal value?

D$)$ If there is no condition on $c,$ for all $i=1,$dots,$n,$ then does the LP in $(2)$ have an optimal solution? If

yes, what is it and what is the optimal value?

E$)$ Is the feasible set of $(2)$ bounded?