Mark the correct alternative in the following:

Formula:- The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$

Given:-

$f(x)=x^{x}$

$\mathrm{d}\left(\frac{\mathrm{f}(\mathrm{x})}{\mathrm{dx}}\right)=\mathrm{x}^{\mathrm{x}}(1+\log \mathrm{x})=\mathrm{f}(\mathrm{x})$

now for decreasing

$f^{\prime}(x)<0$

$\Rightarrow x^{x}(1+\log x)<0$

$\Rightarrow(1+\log x)<0$

$\Rightarrow \log x<-1$

$\Rightarrow x

$x \in\left(0, \frac{1}{e}\right)$