Question

All possible values of $x$ for which the function
$f\left(x\right)=x\ln x-x+1$ is positive is

• A. $(1,\infty )$
• B. $(\frac{1}{e},\infty )$
• C. $[e,\infty )$
• D. $(0,\infty )-\left\{1\right\}$

Answer 1

Answer: (D)

$(0,\infty )-\left\{1\right\}$

The function is:

$f\left(x\right)=x\ln x-x+1$

Domain is $x>0$ for the ln function to exist.

$\Rightarrow f^{\prime }\left(x\right)=x\times \frac{1}{x}+\ln x-1$

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

For f'(x) to exist $x>0$

Intersection of both the domains give for the function to be positive $x>0$

Only at $x=0$ we have $f\left(x\right)=0$ hence the The total domain for the function to be positive is $(0,\infty )-1$. ....Answer