Let $f : \left[ \frac { 1 } { 2 } , 1 \right] \rightarrow \mathbb { R }$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f ^ { \prime } ( x ) < 2 f ( x )$ and $f \left( \frac { 1 } { 2 } \right) = 1$. Then the value of $\int _ { 1/2 } ^ { 1 } f ( x ) d x$ lies in the interval\\
(A) $( 2 e - 1,2 e )$\\
(B) $( e - 1,2 e - 1 )$\\
(C) $\left( \frac { e - 1 } { 2 } , e - 1 \right)$\\
(D) $\left( 0 , \frac { e - 1 } { 2 } \right)$