Please do not use AI to solve. I have tried bard and GPT$4$ and both give incorrect answers.

The structure you are designing contains a cantilever beam of rectangular cross$-$section similar

to Problem $4,$ but there is no limit stop. Under the load $F=0\mathrm{.}001N,$ the free end needs to

deflect $0\mathrm{.}25mm.b$ and $h$ are free design variables, but must be kept within the following ranges:

$.$ Select the values of $b$ and $h$ that maximize the

safety of factor of the beam.

The beam is made of polysilicon $($a brittle material$)$ and has the following properties:

$\{$:$160$GPa,$S_{ut}=1\mathrm{.}4$GPa,$S_{uc}=20$GPa$).$ Use the Brittle Coulomb$-$Mohr failure theory.