Let $f : \mathbf { R } \rightarrow \mathbf { R }$ be a twice differentiable function such that $f ( x + y ) = f ( x ) f ( y )$ for all $x , y \in \mathbf { R }$. If $f ^ { \prime } ( 0 ) = 4 \mathrm { a }$ and $f$ satisfies $f ^ { \prime \prime } ( x ) - 3 \mathrm { a } f ^ { \prime } ( x ) - f ( x ) = 0 , \mathrm { a } > 0$, then the area of the region $\mathrm { R } = \{ ( x , y ) \mid 0 \leq y \leq f ( \mathrm { a } x ) , 0 \leq x \leq 2 \}$ is: (1) $e ^ { 2 } - 1$ (2) $\mathrm { e } ^ { 2 } + 1$ (3) $e ^ { 4 } + 1$ (4) $e ^ { 4 } - 1$
Let $f : \mathbf { R } \rightarrow \mathbf { R }$ be a twice differentiable function such that $f ( x + y ) = f ( x ) f ( y )$ for all $x , y \in \mathbf { R }$. If $f ^ { \prime } ( 0 ) = 4 \mathrm { a }$ and $f$ satisfies $f ^ { \prime \prime } ( x ) - 3 \mathrm { a } f ^ { \prime } ( x ) - f ( x ) = 0 , \mathrm { a } > 0$, then the area of the region $\mathrm { R } = \{ ( x , y ) \mid 0 \leq y \leq f ( \mathrm { a } x ) , 0 \leq x \leq 2 \}$ is:\\
(1) $e ^ { 2 } - 1$\\
(2) $\mathrm { e } ^ { 2 } + 1$\\
(3) $e ^ { 4 } + 1$\\
(4) $e ^ { 4 } - 1$