QPaperGen

Let \(x_1\) and \(x_2\) be any two real numbers such that \(x_1 < x_2\).

Then:
\[x_1 < x_2 \Rightarrow 2x_1 < 2x_2\]
\[\Rightarrow e^{2x_1} < e^{2x_2}\]
\[\Rightarrow f(x_1) < f(x_2)\]


Hence, \(f(x)\) is strictly increasing on \(\mathbb{R}\).
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\(\bf Alternate Method:\) Differentiating w.r.x. we get
\[f'(x) =2 e^{2x}\]
It is always positive for any value of x. So \(f(x)\) is strictly increasing on \(\mathbb{R}\).