Question

The greatest & least value of $f\left(x\right)=x^2\log x$ in the interval $[1,e]$ are respectively

• A. $-\frac{1}{2e},0$
• B. $0,-\frac{1}{2}$
• C. $0,-\frac{1}{4}$
• D. $e,\frac{1}{e}$

Answer 1

The correct option is A $-\frac{1}{2e},0$

Given, $f\left(x\right)=x^2\log x$

For calculculating greatest and least value we have to calculate $f^{\prime }\left(x\right)$
$f^{\prime }\left(x\right)=2xlogx+x=0$

$f^{\prime }\left(x\right)=0\Rightarrow x=0,e^{\frac{-1}{2}}$
$f\left(0\right)=0$ (least value)
$f\left(e^{\frac{-1}{2}}\right)=\frac{-1}{2e}$ (greatest value)