Mark the correct alternative in the following:

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

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

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