Problem $4$: You are given the following tableau for a maximization problem.

Give conditions on the unknowns $a_1,a_2,a_3,b,$ and $c$ that make the following statements true. $($That is$,$

your conditions should specify all of the cases under which the following statements are true.$)$

a$.$ The current solution is optimal.

b$.$ The current solution is optimal, but there are alternative optimal solutions.

c$.$ The current basic solution is feasible, but the LP is unbounded.

d$.$ The current basic solution is a degenerate basic feasible solution.

e$.$ The current basic solution is feasible, but the objective can be improved by replacing $x_5$ as a

basic variable with $x_1.$

Problem $5$: In problem $4,$ suppose the unknowns take the following values:

$a_1=2,a_2=-1,a_3=-4,b=0,c=16$

For this problem, how many optimal solutions are there? Explain your answer. Before answering this

question, note that the ratio test fails if we enter $x_1.($Hint: Let the entering variable take a positive

value while holding all other nonbasic variables equal to zero. Then figure out how the basic variable

values change.$)$