Let $f ( x )$ be a non-constant twice differentiable function defined on $( - \infty , \infty )$ such that $f ( x ) = f ( 1 - x )$ and $f ^ { \prime } \left( \frac { 1 } { 4 } \right) = 0$. Then,\\
(A) $f ^ { \prime \prime } ( x )$ vanishes at least twice on $[ 0,1 ]$\\
(B) $f ^ { \prime } \left( \frac { 1 } { 2 } \right) = 0$\\
(C) $\quad \int _ { - 1 / 2 } ^ { 1 / 2 } f \left( x + \frac { 1 } { 2 } \right) \sin x d x = 0$\\
(D) $\int _ { 0 } ^ { 1 / 2 } f ( t ) e ^ { \sin \pi t } d t = \int _ { 1 / 2 } ^ { 1 } f ( 1 - t ) e ^ { \sin \pi t } d t$