Let $$f : \left[\frac{1}{2}, 1\right] \rightarrow R$$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $$f'(x) < 2 f (x)$$ and $$f\left(\frac{1}{2}\right) = 1$$ Then the value of $$\int_{\frac{1}{2}}^{1} f(x) dx$$ lies in the interval

A

$$(2 e - 1, 2e)$$

B

$$(e - 1, 2e - 1)$$

C

$$\left(\frac{e - 1}{2}, e - 1 \right)$$

D

$$\left(0, \frac{e - 1}{2}\right)$$