Question $5.$ Consider the following linear program:

$mamize3x_1+x_2$

subject $to5x_1+2x_{2}20$

$5x_1+x_{2}15$

$x_1,x_{2}0.$

a$)$ Solve the problem using the graphical approach. Show feasible region and optimal solution. Show all

work.

b$)$ How much can the objective function coefficient of $x_1$ decrease before the optimal solution from a$)$ is

no longer optimal?

c$)$ How much can the objective function coefficient of $x_1$ increase before the optimal solution from a$)$ is

no longer optimal?

d$)$ We saw in class that changing the right hand side of a binding constraint by even a small amount may

change the optimal objective function value.

i$.$ What is the change per unit in the optimal objective function value if we increase the right hand

side of the first constraint by $1$ unit?

ii$.$ What is the change per unit in the optimal objective function value if we increase the right hand

side of the first constraint by $2$ units?

iii. What is the change per unit in the optimal objective function value if we decrease the right hand

side of the first constraint by $1$ unit?

e$)$ What is the change per unit in the optimal objective function value if we increase the right hand side

of the first constraint by a sufficiently small amount $?$