Bonus Problem $(50$ pts$)$

The free end $B$ of a rectangular cross$-$sectioned cantilever beam $AB$ is deflected by $$ under a concentrated load $P.$ The beam has a length $L,$ Young's modulus $E,$ and density $($mass per unit volume$).$ The height $h$ of the beam cross$-$section is uniform, while the width $b$ varies along the beam axis, i$.$e$.,b=b(x).$ Gravity is neglected in this problem.

$(1)$ Please show that the relation below holds, if the beam width is uniform, i$.$e$.,b(x)-=b_0,$

$=\frac{P}{E}{\displaystyle \underset{0}{\overset{L}{}}}\frac{(x-L{)}^2}{\frac{1}{12}{b}_0{h}^3}dx.$

Please prove that the relation below holds, even if the beam width $b(x)$ is nonuniform,

$=\frac{P}{E}{\displaystyle \underset{0}{\overset{L}{}}}\frac{(x-L{)}^2}{\frac{1}{12}b(x)h^3}dx$

$(3)$ Assuming the total mass of the beam is fixed at $m,$ please determine $($or guess $(2))$ the optimal width profile $b(x)$ that minimizes the deflection magnitude $.($All the other quantities, including $P,L,h,E,,$ are fixed.$)$

$4$