Consider the following LP model:

Maximize $z=4x_1+2x_2$

subject $to3x_1+2x_{2}12$

$x_1+x_{2}5$

$x_1,x_{2}0$

By using graphical sensitivity analysis, answer the following questions.

$($a$)$ Find the dual prices of constraints $1$ and $2.$

$($b$)$ Find the feasibility ranges of the constraints $1$ and $2.$

$($c$)$ Find the optimality ranges of the objective function coefficients.

$($d$)$ If the right$-$hand side of constraint $1$ is decreased to $10,$ how will this decrease impact the

optimum solution.

$($e$)$ Suppose that the objective coefficient of $x_1$ is increased to $6.$ Will the current optimum

remain the same?