Let f: ℝ → ℝ be a twice differentiable function such that f(x + y… | JEE Main 2025 Mathematics PYQ

MCQMediumJEE Main 2025 · 22 January, Shift 1Linear Differential Equations

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 $f''(x) - 3a f'(x) - f(x) = 0$ , where $a > 0$ , then the area of the region $R = \{(x, y) | 0 \leq y \leq f(ax), 0 \leq x \leq 2\}$ is :

A
$e^2 - 1$
B
$e^4 + 1$
C
$e^4 - 1$
D
$e^2 + 1$

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