Correct option is (4) \(\left[\frac{1}{\mathrm{e}}, \infty\right)\)
\(\mathrm{f}(\mathrm{x})=\mathrm{x}^{\mathrm{x}} ; \mathrm{x}>0\)
lny = xlnx
\(\frac{1}{y} \frac{d y}{d x}=\frac{x}{x}+\ell n x\)
\(\frac{d y}{d x}=x^{x}(1+\ell n x)\)
for strictly increasing
\(\frac{\mathrm{dy}}{\mathrm{dx}} \geq 0 \Rightarrow \mathrm{x}^{\mathrm{x}}(1+\ell \mathrm{n} x) \geq 0\)
\(\Rightarrow \ell \mathrm{nx} \geq-1\)
\(\mathrm{x} \geq \mathrm{e}^{-1}\)
\(\mathrm{x} \geq \frac{1}{\mathrm{e}}\)
\(\mathrm{x} \in\left[\frac{1}{\mathrm{e}}, \infty\right)\)
