Show that the function given by $f\left( x \right) = {e^{2x}}$ is strictly increasing on $R$.

We have, $f(x) = {e^{2x}}$ …(i)
$f(x)$ being an exponential function, is continuous and derivable on $R$.

Differentiating (i) w.r.t. $x$, we get
$f(x) = {e^{2x}} \cdot 2 = 2P > 0$ for all $x \in R$
$\Rightarrow f$ is strictly increasing on $R$.