question_answer

Let \[f:R\to R\] be a continuous odd function, which vanishes exactly at one point and \[f(1)=\frac{1}{2}\]. Suppose that \[F(x)=\int\limits_{-1}^{x}{f(t)}\,\,dt\] for all \[x\in [-1,\,\,2]\] and \[G\,\,(x)=\int\limits_{-1}^{x}{t|\{f\,\,(t)\}|}\,\,dt\] for all \[x\in [-1,\,\,2]\]. If \[\underset{x\to 1}{\mathop{\lim }}\,\frac{F(x)}{G(x)}=\frac{1}{14},\] then the value of \[f\,\,\left( \frac{1}{2} \right)\] is _______.

A)  1                                

B)  7

C)  3                                

D)  9

E)  None of these