Q.Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(x + y) = f(x)f(y)$ for all $x, y \in \mathbb{R}.$ If $f'(0) = 4a$ and $f$ satisfies $f''(x) - 3a f'(x) - f(x) = 0, a > 0,$ then the area of the region $R = \{(x,y) | 0 \le y \le f(ax), 0 \le x \le 2\}$ is:

e^{2} - 1

e^{4} + 1

e^{4} - 1

e^{2} + 1

The correct solution involves applying the fundamental concept to derive the final value step by step...

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