Question

All possible value of $x$ for which the function $f\left(x\right)=\frac{x}{Inx}-x+1$ is positive is

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

Answer 1

Answer: (A)

$(1,\infty )$

$f\left(x\right)=\frac{x}{\ln \left(x\right)}-x+1>0$

Existance of $\ln x\Rightarrow x>0$ and $x\ne 1$

For $x\in (0,1)$

$f\left(x\right)=\frac{(0,1)}{(-\infty ,0)}-(0,1)+1\ngtr 0$

Now, for $x\in (1,\infty ),f\left(x\right)=\frac{(1,\infty )}{(0,\infty )}-(1,\infty )+1>0$

$\Rightarrow x\in (1,\infty )$