If $f : \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f'(x) > 2f(x)$ for all $x \in \mathbb{R}$, and $f(0) = 1$, then

$\text{Given: } f'(x) > 2f(x)\ \ \forall x \in \mathbb{R}$

$f(0) = 1$

$\text{To find: the correct statement among (1)--(4)}$

$\text{Let } g(x) = f(x)\,e^{-2x}$

$g'(x) = e^{-2x}\big(f'(x) - 2f(x)\big) > 0$

$g \text{ strictly increasing}, \qquad g(0) = f(0)e^{0} = 1$

$x > 0 \Rightarrow g(x) > g(0) = 1$

$\Rightarrow f(x)e^{-2x} > 1 \Rightarrow f(x) > e^{2x}$

$\Rightarrow (3)\ \text{TRUE}$

$f(x) > e^{2x} > 0 \Rightarrow f'(x) > 2f(x) > 0 \Rightarrow (1)\ \text{TRUE}$

$f'(x) > 2f(x) > 2e^{2x} > e^{2x} \Rightarrow (4)\ \text{false}$

$f \text{ increasing} \Rightarrow (2)\ \text{false}$

$\text{TRUE: } (3)\ f(x) > e^{2x}, \text{ and } (1)\ f \text{ increasing}$

$\boxed{(1)\ \text{and}\ (3)}$