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

(a) $f(x)$ is increasing in $(0, \infty)$
(b) $f(x)$ is decreasing in $(0, \infty)$
(c) $f(x)>e^{2 x}$ in $(0, \infty)$
(d) $f'(x)<e^{2 x}$ in $(0, \infty)$

(a), (c)

We have,

$f'(x)>2 f(x) $ for all $x \in R$

$\Rightarrow f'(x)-2 f(x)>0 $ for all $x \in R$

$\Rightarrow f'(x) e^{-2 x}-2 e^{-2 x} f(x)>0 $ for all $x \in R$

$\Rightarrow \frac{d}{d x}\left(f(x) e^{-2 x}\right)>0$ for all $x \in R$

$\Rightarrow f(x) e^{-2 x}$ is an increasing on R

$\Rightarrow f(x) e^{-2 x}>f(0) e^0$ for all x > 0

$\Rightarrow f(x)>e^{2 x}$ for all x > 0

Now,

$f'(x) >2 f(x)$                 [Given]

$\Rightarrow f'(x) >2 e^{2 x}$                   [∵ f(x) > e^2x]

⇒ f'(x) > 0 for all x > 0

⇒ f(x) is increasing in $(0, \infty)$

Hence, options (a) and (c) are true.