Question

Find the greatest value of $f\left(x\right)=x^2\log x$ on $[1,e];$

• A. $0$
• B. $e^2$
• C. $e/2$
• D. $1$

Answer 1

The correct option is B $e^2$

Given $f\left(x\right)=x^2\log x$
$f^{\prime }\left(x\right)=x(1+2\log x)=0$ for max. or min.
$\Rightarrow x=0$ or $x=e^{-1/2}.$ Since $0<e^{-1/2}<1,$ none of these critical points lies in the interval $[1,e].$
So we only compute the values of $f\left(x\right)$ at the end points $1$and $e$
We have $f\left(1\right)=0,f\left(e\right)=e^2.$
Also in the given interval $f^{\prime }\left(x\right)>0$, so $f\left(x\right)$ is increasing in the given interval
Thus $f\left(1\right)=0$ is the least value and $f\left(e\right)=e^2$ the greatest value of the function.