Given the linear programming problem:

Minimize z $=$x$1$ x$33$x$4$

subject to

$3$x$1+2$x$23$x$3$ x$4>=3$

x$13$x$22$x$32$x$4>=6$

x$1,$ x$2,$ x$3,$ x$4>=0$

$($a$)$ Construct the dual problem.

$($b$)$ Use the Strong Duality Theorem to show that

x$=$

$$

$$

$$

$$

$0$

$0$

$0$

$3$

$$

$$

$$

$$ and y$=$

$$

$$

$5$

$7$

$8$

$7$

$$

$$ are optimal feasible solutions of the

primal and dual problems, respectively.

$($c$)$ Verify the complementary slackness condition