Q$3.[25$ points$]$ A linear elastic cantilever of one meter long is subjected to a downward vertical force $\backslash (\backslash$mathrm$\{$P$\}=2\backslash$mathrm$\{$kN$\}\backslash )$ at its tip. The geometry of the cross section of the beam $($front view$)$ is given in Figure $3.$ Assume that no twisting occurs. $\backslash ($ E$=165\backslash$mathrm$\{$GPa$\}\backslash ).$

Determine

$($i$)$ the location of the centroid,

$($ii$)$ the moments of inertia of the cross section Iy$,$ Iz and Iyz, with z and y being axes along horizontal $($to right$)$ and vertical $($down$)$ directions and passing the centroid.

$($iii$)$ the equation of the neutral axis and its direction, and show it in the figure.

$($iv$)$ the maximum tensile and the maximum compressive bending stresses at the root section.