For the beam and loading shown in the figure,

a$)$ Draw the bending moment and shear force diagrams;

b$)$ Solve the thickness $\backslash (($h$)\backslash )$ of the beam, knowing that the timber has an allowable normal stress of $12$ MPa $.$

c$)$ Derive the deflection of the beam AC using the forth$-$order differential equation.

d$)$ Determine the vertical displacement $\backslash ((\backslash$delta$)\backslash )$ and rotation angle $\backslash ((\backslash$theta$)\backslash )$ at point C $.$