$(20$ points$)$ Consider the linear programming problem

minimize $z=10x_1-2x_2+5x_3$

subject $to{x}_1+x_2-2x_{3}1$

$2x_1+x_{3}4$

$-x_2+x_{3}0$

$x_1,x_2,x_{3}0$

$($i$)$ Show that $x=(2,1,1{)}^T$ is a feasible point and label each of the constraints as binding $($active$)$ or

non$-$binding $($inactive$).($Do not forget the non$-$negativity constraints$)$

$($ii$)$ Find the set of all feasible directions $p=(p_1,p_2,p_3{)}^T$ at $x.$

$($iii$)p=(-1,-1,-1{)}^T$ is a feasible direction. Determine the maximal step $>0$ such that $x+p$ remains

feasible.