Answer
The wavelength shift for Compton scattering is maximum when incident photon completely reverses direction and electron goes forward with maximum kinetic energy. In this case scattering angle $\theta=180^{0}$.
Work Step by Step
Wavelength shift or Compton shift is given by
$\Delta \lambda= \lambda- \lambda_{0}=\lambda_{C} (1-cos\theta)$
As value of $cos\theta $ is always between +1 and -1, maximum $\Delta \lambda$ will occur only when $cos\theta =-1$ or $\theta =180^{0}$.
in this case
$\Delta \lambda= \lambda- \lambda_{0}=\lambda_{C} (1-cos\theta)=\lambda_{C} (1-cos180^{0})$
$\Delta \lambda= \lambda- \lambda_{0}=\lambda_{C} (1-(-1))=2\lambda_{C}$
