Consider the problem of minimizing c$$xc$\text{'}$xc$$x over a polyhedron PPP$.$ Prove the following:

$($a$)(5$ points$)$ A feasible solution xxx is optimal if and only if c$$d$0$c$\text{'}$d $\backslash$geq $0$c$$d$0$ for every feasible direction ddd at xxx$.$

$($b$)(5$ points$)$ A feasible solution xxx is the unique optimal solution if and only if c$$d$>0$c$\text{'}$d $>0$c$$d$>0$ for every nonzero feasible direction ddd at xxx$.$