A simply supported beam carries a point load $(P)$ as shown in Figure $1.$ The deflection, y$,$ caused by the load along the beam is governed by the following equation:

$y=\frac{Pbx}{6\text{E}\text{I}\text{L}}(L^2-b^2-x^2),$ for $0x(L-b)$

where $P,b,E,$I and L are positive constant parameters.

Figure $1$

$($a$)$ Determine the value of x where the value of y is maximum. Expressed your answer in terms of P$,$ b$,$ E$,$ I and L$.$

$($b$)$ Determine the maximum value of y$.$ Express your answer in terms of $P,b,E,$I and L $.$

$($c$)$ Consider the deflection curve:

$y=\frac{Pbx}{6\text{E}\text{I}\text{L}}(L^2-b^2-x^2),$ from $0to(L-b)$

$($d$)$ Prove that the area under the curve is:

$\frac{Pb}{24\text{E}\text{I}\text{L}}(L-b{)}^3(L+3b)$