$\begin{aligned}
& \mathrm{f}(\mathrm{x})=\mathrm{x}^{\mathrm{x}} ; \mathrm{x}>0 \
& \ell \operatorname{nn}=\mathrm{x} \ell \mathrm{n} \mathrm{x} \
& \frac{1}{\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}}{\mathrm{x}}+\ln \mathrm{n} \
& \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x}^{\mathrm{x}}(1+\ell \mathrm{n} x)
\end{aligned}forstrictlyincreasing\begin{aligned}
& \frac{d y}{d x} \geq 0 \Rightarrow x^x(1+\ell n x) \geq 0 \
& \Rightarrow \ell n x \geq-1
\end{aligned}\begin{aligned} & x \geq e^{-1} \ & x \geq \frac{1}{e} \ & x \in\left[\frac{1}{e}, \infty\right)\end{aligned}$