$($b$)$ A simply supported beam carrying a concentrated load and a uniformly distributed load is shown below in Figure Q$1$b$.$

$($i$)$ Derive the shear force and bending moment functions for the beam.

$($ii$)$ Draw the shear force and bending moment diagrams.

$($iii$)$ Determine the position and magnitude of the maximum bending moment. The mass of the beam may be neglected.

$(12$ marks$)$