Question

The set onto which the derivative of the function $f\left(x\right)=x(\log x-1)$ maps the range $[1,\infty )$ is

• A. $[1,\infty )$
• B. $(e,\infty )$
• C. $[e,\infty )$
• D. $(0,0)$

Answer 1

The correct option is C $[e,\infty )$

Given $f\left(x\right)=x[\log x-1]x\in [+,\infty ]$

$f\left(x\right)=x\left[\frac{d}{dx}\right(\log x-1\left)\right]+(\log x-1)\frac{d}{dx}\left(x\right)$

$=x.\frac{1}{x}+(\log x-1).1$

$=1-1\log x-1$

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

$\therefore$ The set of $f\left(x\right)=\log x$ that maps

$[1,\infty )$ is $[e,\infty )$