Let f: R → R be a twice differentiable function such that f(x+y)=f(x) f(y) for all x, y ∈ R. If f^′(0)=4 a and f satisfies f^′ ′(x)-3 a…

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