$1.(16$ marks$)$ An overhanging beam is subjected to the loads shown below in Fig. $($a$)$ and its cross section is demonstrated in Fig. $($b$).$ Determine $($i$)$ the functions of the internal bending moment and plot the diagram of bending moment; $($ii$)$ the maximum tensile bending stress and the maximum compressive bending stress; $($iii$)$ the minimum bending stress and where it occurs.

$($a$)$