Question

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

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

Answer 1

Answer: (B)

$0,e^2$

$f\left(x\right)=x^2\log x$
$f^{\prime }\left(x\right)=x(1+2\log x)$
Now, $f^{\prime }\left(x\right)>0$ in the interval $[1,e]$ since both the terms $x$ and $(1+2\log x)$ are positive in that interval.
Hence, the function is increasing in the interval $[1,e].$
So, the smallest value of the function in that interval will be at $x=1$ and the greatest value will be at $x=e.$
Hence, least and greatest values of $f\left(x\right)$ are $0,e^2$