Consider the following linear programming problem:

Minimize $4$A $+5$B

Subject to

$1$A $+4$B $<=21$

$2$A $+1$B $>=7$

$3$A $+1\mathrm{.}5$B $<=21$

$2$A $+6$B $>=0$

A$,$ B $>=0$

$($a$)$ Show the feasible region using the graphical solution approach. What are the corner points?

$($b$)$ Based on your response in part $($a$)$:

What is the corner point that minimizes the objective function?

What is the value of the objective function at this corner point?

$($c$)$ Which constraints are binding? Which constraints are non$-$binding? Explain.

$($d$)$ Suppose the objective function is changed to Maximize $7$A $+5$B$,$ while keeping everything else intact.

What is the corner point that maximizes the new objective function?

What is the value of the objective function at this corner point?

$($e$)$ Taking into consideration the new objective function introduced in part $($e$),$ which constraints are binding? Which constraints are non$-$binding? Explain.