JEE Main 2024 — Application of Derivatives Question with Solution

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