Minimize Z $=2$x$1+$ x$2($Note that you are minimizing$)$ subject to x$1+$ x$2=8($note the equal sign$)$

x$10,$ x$20$

a$)(5$ pts$)$ Using any method you want indicate the optimal solution.

Next, we consider using the interior point method to solve this problem with initial trial solution $($x$1,$ x$2)=(4,4).$

b$)(10$ pts$)$ Give the result of first iteration for $=0\mathrm{.}5?$

c$)(10$ pts$)$ Give the result of second iteration for $=0\mathrm{.}5?$

d$)(10$ pts$)$ Give the result of first iteration for $=0\mathrm{.}75?$

e$)(5$ pts$)$ Explain what happens in this problem if we set $=1$ at any step in the iterative process?