The forces acting on the system are shown in the diagram

For vertical equilibrium of the point $\mathrm{P}$

$T = F$                      ....(i)



And for the vertical equilibrium of the system



$R + T = (m+M)g$,  i.e.,  $R = (m+M)g - T$             ....(ii)

Now the system will not move horizontally till

              $T < f_{\mathrm{L}}$

i.e.,        $T < \mu [\left(m+M\right)g - T]$              $\left( f_{\mathrm{L}} = \mu R\right)$

which on simplification gives

$T<\frac{\mu \left(\mathit{\text{m}}+\mathit{\text{M}}\right)\mathit{\text{g}}}{\left(1+\mu \right)}$

So in the light of equation (i), we get

$F_{\text{max}}=\frac{\mu \left(\mathit{\text{m}}+\mathit{\text{M}}\right)\mathit{\text{g}}}{\left(1+\mu \right)}$