Question

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

• A. $[1,\infty )$
• B. $(10,\infty )$
• C. $[0,\infty )$
• D. $(0,0)$

Answer 1

Answer: (C)

$[0,\infty )$

$f\left(x\right)=x(\log x-1)$

$f^{\prime }\left(x\right)=x(\frac{1}{x}-0)+(\log x-1)\cdot 1$ (Leibniz rule)

$=1+\log x-1=\log x$

Domain given $[1,\infty )$

$\Rightarrow \log 1=0$

$\log \infty =\infty$

Range is $[0,\infty )$