Problem $(30$ pts$).$ Consider the following LP problem $($a modification on Wyndor Glass problem$).$ Maximize Z $=3$ x$1+5$x$2,$ subject to x$142$x$2123$x$1+2$x$2=18($updated constraint$)$ And x$10,$ x$20$ a$)(5$ pts$)$ Write down the complete artificial problem $($in augmented form$)$ to be used by the Big M method. Don$$t forget about the function to be optimized. b$)(5$ pts$.)$ Using any method you want, identify the optimal solution. c$)(5$ pts$)$ Suppose that you forgot to modify the objective function to account for artificial variables. Explain what would happen to the optimal solution in this particular case. $($Using any method you want to justify your answer is fine. You don$$t need to apply the big M method.$)$ d$)(5$ pts$)$ Suppose that objective function above changed to minimize Z $=3$ x$1+5$x$2,$ subject to the same boundary constraints as above. Write down the complete artificial problem $($in augmented form$)$ to be used by the Big M method. Don$$t forget about the function to be optimized. e$)(5$ pts$.)$ Using any method you want, identify the optimal solution to problem in part $($d$).$ f$)(5$ pts$)$ Suppose that you forgot to modify the objective function to account for artificial variables in part d$.$ Explain what would happen to the optimal solution in this particular case. $($Using any method you want to justify your answer is fine. You don$$t need to apply the big M method.$)$