$($Maximum $10$ points$)$ Consider the fixed$-$fixed beam $AB$ of length $L$ shown in figure $1.$ A load $P$ acts at the mid span. Neglect any axial deformation.

Figure $1$: Beam for Question $1.$

$($a$)(5$ points$)$ Determine the fixed end moments developed at the supports A and $B.$

$($b$)(5$ points$)$ Utilizing the fixed end moments calculated in the previous part, find the maximum deflection in the beam using conjugate beam method.