Consider the following linear program.

$]6[0$

$($a$)$ Use the graphical solution procedure to find the optimal solution.

What is the value of the objective function at the optimal solution?

$at(x,y)=(,)$

$($b$)$ Assume that the objective function coefficient for $x$ changes from $8$ to $6.$ Use the graphical solution procedure to find the new optimal solution.

Does the optimal solution change?

The extreme point optimal. The value of the objective function becomes

$($c$)$ Assume that the objective function coefficient for $x$ remains $8,$ but the objective function coefficient for $Y$ changes from $12$ to $6.$ Use the graphical solution procedure to find the new optimal solution.

Does the optimal solution change?

The extreme point optimal. The value of the objective function becomes

$($d$)$ The computer solution for the linear program in part $($a$)$ provides the following objective coefficient range information.