Question

Difference between the greatest and the least values of the function $\left(x\right)=x(Inx-2)$ on $[1,e^2]$ is

• A. $2$
• B. $e$
• C. $e^2$
• D. $1$

Answer 1

The correct option is A $2$

$f\left(x\right)=x(lnx-2)$

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

$f^{\prime }\left(x\right)=x\frac{1}{x}+lnx-1$

$f^{\prime }\left(x\right)=lnx$

$f^{\prime }\left(x\right)=0$

$lnx=0$

$x=1$

$x>1$, $f^{\prime }\left(x\right)>0$

$\therefore f\left(x\right)$ is an increasing function $f\left(x\right)$ has least value at $x=1$ & greatest value at $e^2$ in the interval $[1,e^2]$

$f\left(1\right)=-2$

$f\left(e^2\right)=e^2(ln{e}^2-2)$

$=e^2(2lne-2)$

$=0$

$f\left(e^2\right)=f\left(1\right)=0-(-2)=2$

$\therefore$ Answer: $2$.