Question

On $[1,e]$, the least and greatest values of $f\left(x\right)=x^2$ ln x is

• A. $e,1$
• B. $1,e$
• C. $0,e^2$
• D. None of these

Answer 1

The correct option is C $0,e^2$

$f\left(x\right)=x^2\ln x$

$f^{\prime }\left(x\right)=\frac{x^2}{x}+2x\ln x$

$f^{\prime }\left(x\right)=x+2x\ln x$

$=x(1+2\ln x)x\in [1,e]$

$1+2\ln x=0$

$x=\ln x=-1/2$

$x=e^{-1/2}$

$x=\frac{1}{\sqrt{e}}$

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

strictly increasing function

$f(x{)}_{max}atx=e$

$f\left(x\right)=e^2$

$f(x{)}_{min}$at $x=1$

$f\left(x\right)=0$

Range $\in [0,e^2]$