Consider the following linear program.

$[$ Min $8$ X$+12$ Y; s$.$t$.$; $[1$ X$+3$ Y $>=7$; $2$ X$+2$ Y $>=10$; $6$ X$+2$ Y $>=14$; X$,$ Y $>=0]]$

$($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 $($X$,$ Y$)=($ optimal. The value of the objective function becomes new optimal solution. Does the optimal solution change? The extreme point $($X$,$ Y$)=()$ Select$$ V optimal. The value of the objective function becomes$($c$)$ Assume that the objective function coefficient for X remains $8,$ but the objective function coefficient for Y changes from $12$ to $6.$ Use the graphical solution procedure to find the new optimal solution. Does the optimal solution change? The extreme point $($X$,$ Y$)=($ optimal. The value of the objective function becomes $($d$)$ The computer solution for the linear program in part $($a$)$ provides the following objective coefficient range information. How would this objective coefficient range information help you answer parts $($b$)$ and $($c$)$ prior to re$-$solving the problem? The objective coefficient range for variable X is $$ Select$$ change. The objective coefficient range for variable Y is optimal solution Since the change in part $($b$)$ is $$Select$$ this range, we know the optimal solution to Since the change in part $($c$)$ is this range, we know the change.