A primal linear program has constraints

$2$x$1+$ x$2<=10,$

x$1+2$x$2<=10,$x$1+2$x$2<=6,$ x$1,$x$2>=0.$

Figure $1$ illustrates the feasible region.

Determine whether the following statements are true or false and justify your answer.

$($a$)$

$($b$)$

One of many possible optimal solutions is $($x$1,$x$2)=(2,2),$ but we cannot be sure if we do not know the objective func$-$ tion.

If the optimal solution of the linear program is at $($x$1,$ x$2)=(0,0),$ then the optimal solution of the dual also has all deci$-$

sion variables equal to zero.

$($c$)$ For one given objective, the optimal solution could be at

$($x$1,$ x$2)=(2,4)$ and at $($x$1,$ x$2)=(10/3,10/3).$

$($d$)$ Depending on the objective, this linear program could be infeasible.