Question

The function $x^2\log x$ in the interval $(1,e)$ has

• A. A point of maximum
• B. A point of minimum
• C. Point of maximum as well as of a minimum
• D. Neither a point of maximum nor minimum

Answer 1

The correct option is D

Neither a point of maximum nor minimum

The explanation for the correct answer.

Solve for the points of extremum of a function $x^2\log x$

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

$f\text{'}\left(x\right)=2x\log x+x$ ;($\therefore \frac{d}{dx}(u\times v)=u\times \frac{dv}{dx}+v\times \frac{du}{dx}$)

Now finding the second derivative.

$f\text{'}\text{'}\left(x\right)=2(1+\log x)+1$

$f\text{'}\text{'}\left(1\right)=3+2{\log }_{e}1$

$f\text{'}\text{'}\left(e\right)=3+2{\log }_{e}e$

$f\left(x\right)$ has a local minimum at $\frac{1}{\sqrt{e}}$, but lies only in the interval $(1,e)$so that$f\left(x\right)$ has no maximum and minimum in $(1,e)$

So there is neither a point of maximum nor minimum.

Hence, option(D) is the correct answer.