Mathematics - Application of Derivative Question with Solution | TestHub

$\begin{array}{rl}& \mathrm{f}(\mathrm{x})={\mathrm{x}}^{\mathrm{x}};\mathrm{x}>0\\ & \ell \mathrm{nn}=\mathrm{x}\ell \mathrm{nx}\\ & \frac{1}{\mathrm{y}}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}}{\mathrm{x}}+\ln \mathrm{n}\\ & \frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{x}}^{\mathrm{x}}(1+\ell \mathrm{n}x)\end{array}$ for strictly increasing $\begin{array}{rl}& \frac{dy}{dx}\ge 0\Rightarrow x^x(1+\ell nx)\ge 0\\ & \Rightarrow \ell nx\ge -1\end{array}$ $\begin{array}{rl}& x\ge e^{-1}\\ & x\ge \frac{1}{e}\\ & x\in \left[\frac{1}{e},\infty \right)\end{array}$