Question

In the interval $(0,1)$ maximum value of the function $f\left(x\right)=|x\ln x|$ is -

• A. $0$
• B. $1$
• C. $e$
• D. $e^{-1}$

Answer 1

The correct option is D $e^{-1}$

$f\left(x\right)=-x\ln x$

$f^{{}^{\prime }}\left(x\right)=-1-\ln x$
$f^{{}^{\prime }}\left(x\right)=0$ $\Rightarrow x=\frac{1}{e}$
$f^{{}^{\prime }}\left(x\right)>0\text{if}x\in (0,\frac{1}{e})$ & $f^{{}^{\prime }}\left(x\right)<0\text{if}x\in (\frac{1}{e},0)$
$\Rightarrow x=\frac{1}{e}$ is a point of maxima
$f\left(\frac{1}{e}\right)=e^{-1}$