A society has three consumers$-1,2$ and $3-$ whose demand for a given good is $Q_1=5-P,Q_2=10-P$ and $Q_3=15-P$ respectively. Required:

$($a$)$ Using well labelled diagrams demonstrate the optimal provision of this good if it were a:

$($i$)$ A pure private good $\{2$ marks $\}$

$($ii$)$ A pure public good marks