JEE Main 2024 — 6 April, Shift 1 — Question 14

$\mathrm{f}(\mathrm{x})=\mathrm{x}^{\mathrm{x}} ; \mathrm{x}>0$ $\ell n y=x \ln x$

$\frac{1}{y} \frac{d y}{d x}=\frac{x}{x}+\ln x$

$\frac{d y}{d x}=x^{x}(1+\ln x)$

for strictly increasing

$\frac{d y}{d x} \geq 0 \Rightarrow x^{x}(1+\ell n x) \geq 0$

$\Rightarrow \ln \mathrm{x} \geq-1$

$x \geq e^{-1}$

$\mathrm{x} \geq \frac{1}{\mathrm{e}}$

$\mathrm{x} \in\left[\frac{1}{\mathrm{e}}, \infty\right)$