Let $f : ( 0 , \pi ) \rightarrow \mathbb { R }$ be a twice differentiable function such that
$$\lim _ { t \rightarrow x } \frac { f ( x ) \sin t - f ( t ) \sin x } { t - x } = \sin ^ { 2 } x \text { for all } x \in ( 0 , \pi )$$
If $f \left( \frac { \pi } { 6 } \right) = - \frac { \pi } { 12 }$, then which of the following statement(s) is (are) TRUE?
(A) $f \left( \frac { \pi } { 4 } \right) = \frac { \pi } { 4 \sqrt { 2 } }$
(B) $f ( x ) < \frac { x ^ { 4 } } { 6 } - x ^ { 2 }$ for all $x \in ( 0 , \pi )$
(C) There exists $\alpha \in ( 0 , \pi )$ such that $f ^ { \prime } ( \alpha ) = 0$
(D) $f ^ { \prime \prime } \left( \frac { \pi } { 2 } \right) + f \left( \frac { \pi } { 2 } \right) = 0$