The interval in which the function f(x)=x^x, x>0, is strictly inc… - Vidyadip

MCQ

The interval in which the function $f(x)=x^x, x>0,$ is strictly increasing is

A
$\left(0, \frac{1}{ e }\right]$
B
$\left[\frac{1}{ e ^2}, 1\right)$
C
$(0, \infty)$
D
$\left[\frac{1}{ e }, \infty\right)$

✓

Answer

$f(x)=x^x ; x>0$
$\operatorname{lny}=x \ell\ n x$
$\frac{1}{y} \frac{d y}{d x}=\frac{x}{x}+\ln x$
$\frac{d y}{d x}=x^x(1+\ln x)$
for strictly increasing
$\frac{d y}{d x} \geq 0 $
$\Rightarrow x^x(1+\ln x) \geq 0$
$\Rightarrow \ln x \geq-1$
$x \geq e^{-1}$
$x \geq \frac{1}{e}$
$x \in\left[\frac{1}{e}, \infty\right)$

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