Question:
Mark the correct alternative in the following:
The function $f(x)=x^{x}$ decreases on the interval.
A. $(0, \mathrm{e})$
B. $(0,1)$
C. $(0,1 / \mathrm{e})$
D. $(1 / e, e)$
Solution:
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 $x \in\left(0, \frac{1}{e}\right)$