Hint for the first.
With second order Taylor expansion
$f(0)=0=f(\frac 12)-\frac 12f'(\frac 12)+\frac 18f''(c_1)$
$f(1)=0=f(\frac 12)+\frac 12f'(\frac 12)+\frac 18f''(c_2)$
and by substraction
$f'(\frac 12)=\frac 18(f''(c_1)-f''(c_2))$
$\implies |f'(\frac 12)|\leq \frac{2A}{8}$.
In c part..Shouldn't the recoil kinetic energy of electrons be 150+0.25? Since the energy of incident photons is 150 and the resting energy of electrons is $mc^2$ =0.50 MeV, so energy before conservation is 150+0.50 MeV, after conservation energy of scattered photons is $(mc^2)/2$=0.25MeV so recoil kinetic energy= 150+0.50-0.25?