Question $1$a$.(3$ points$)$ A problem with an unbounded feasible region cannot have an optimal solution. True

or false? Explain your answer or provide a counter$-$example.

b$.(3$ points$)$ In each iteration of the simplex method, the value of the objective function strictly

improves. True or false? Explain your answer or provide a counter$-$example.

Answer the following short questions.

a$.(9$ points$)$ Consider the following problem,

maxz$=5x_1+8x_2$

$s.t.x_1+x_{2}5$

$x_1,x_{2}0$

i$.(3$ points$)$ Find an infeasible solution.

ii$.(3$ points$)$ Find an feasible solution.

iii. $(3$ points$)$ Find the optimum solution.