The minimum value of xlog_ex is equal to

The correct answer is option (3) → $-\frac{1}{e}$

$y=x\log x$

so $y'=\log x+1=0⇒x=\frac{1}{e}$

$y''=\frac{1}{x}$ so $(y''(\frac{1}{e})=e)>0$

$\frac{1}{e}$ → point of minima

so $y_{min}=y(\frac{1}{e})=-\frac{1}{e}$