Solve the following linear program using the graphical method and answer all questions. You should submit the picture file of your graphical solution and show your calculations or reasons to answer the questions below. You may consider verifying your solutions and answers by EXCEL $($no need to submit your EXCEL verification$).$

Max $2$x$1+3$x$2$

s$.$t

x$1+$ x$210$

$2$x$1+$ x$26$

x$1+3$x$224$

$3$x$1+$ x$226$

x$1,$ x$20$

$($a$)$ Provide a graphical representation for the above linear program. Identify the optimal solution and the associated optimal objective function value.

$($b$)$ Compute the ranges of c$1$ and c$2($one coefficient is fixed at a time$)$ of the objective function coefficients for x$1$ and x$2,$ for the current optimal solution to remain optimal.

$($c$)$ Suppose c$1$ is increased from $2$ to $2\mathrm{.}5$ and c$2$ is increased from $3$ to $3\mathrm{.}5.$ Will the current optimal solution change?

$($d$)$ Suppose c$1$ is increased from $2$ to $2\mathrm{.}5.$ What is the new optimal solution?

$($e$)$ Suppose c$2$ is decreased from $3$ to $1.$ What is the new optimal solution?

$($f$)$ Compute the dual prices for constraints $1$ and $2$ and interpret them. $($g$)$ Solve the optimal solution by Gurobi. Write a function to get the range of c$1$ and c$2($one coefficient is fixed at a time$)$ for the current optimal solution to remain optimal. You should get the same answer in $($b$).$