$$ \text{Let }f: \mathbb{R} \rightarrow \mathbb{R} \text{ be a twice differentiable function such that } f(x+y)=f(x)f(y)\text{ for all } x,y \in R. \text{ If } f^{'}(0)=4a \text{ and } f \text{ satisfies } f^{''}(x)-3af^{'}(x)=0,a>0, \text{then the area of the region } R= \left\{(x,y) \mid 0\leq y\leq f(ax), 0\leq x \leq2 \right\}$$

A

$$ e^{2}-1$$

B

$$ e^{2}+1$$

C

$$ e^{4}+1$$

D

$$ e^{4}-1$$