$($P$2)($a$)$ Without quoting any theorems from class notes or the text, explain how we know that a standard

form L$.$O$.$P$.$ has either no optimum points, a unique optimum point, or infinitely many optimum

points. Hint: You only need to show that if there is more than one optimal point, then there are

infinitely many. The previous question should help.

$($b$)$ Consider the L$.$O$.$P$.$ below:

Maxz$=3x_1+4x_2$

such that $3x_1+4x_{2}12$

$2x_1+x_{2}6$

and $x_1,x_{2}0.$

Use a tableau and pivot according to the phase II rules of the LS simplex algorithm. What

optimal point to you arrive at$?$ Now show that there is more than one optimal basis $($hence there

is a degeneracy$)$ and parameterize the set of optimum points vec$(x{)}^{*}(t).$