Consider the following linear program.

$]8[0$

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

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

$at(x,n)=(,)$

$($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 $(x,y)=(,)$ remains $v$ 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 $(x,y)=(,)$ becomes $$ 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.

$\backslash$table$[[$Variable$,\backslash$table$[[$Objective$],[$Coefficient$]],\backslash$table$[[$Allowable$],[$Increase$]],\backslash$table$[[$Allowable$],[$Decrease$]]],[x,8\mathrm{.}00000,4\mathrm{.}00000,4\mathrm{.}00000],[Y,12\mathrm{.}00000,12\mathrm{.}00000,4\mathrm{.}00000]]$

How would this objective coefficient range information help you answer parts $($b$)$ and $($c$)$ prior to re$-$solving the problem?

The objective coefficient range for variable $x$ is to

Since the change in part $($b$)$ is within to this range, we know the optimal solution will not $$ change. The objective coefficient range for variable $Y$ is $.$ Since the change in part $($c$)$ is this range, we know the optimal solution $[$ change.