Question

Which one of the following sets is true set of real values of x for which the function, $f\left(x\right)=x\ell nx-x+1$ positive ?

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

Answer 1

Answer: (B)

$(0,1)$

$f\left(x\right)=xlnx-x+1>0$

$xlnx>x-1$

$lnx>\frac{x-1}{x}.$

$case$ $\left(i\right)⟶x>1$

Then $lnx>0$

$\frac{x-1}{x}<0$

$x\in (0,1)$ (Not satisfied).

$case$ $\left(ii\right)$ $0<x<1$

Then $lnx<0$

$\frac{x-1}{x}<0$

$x\in (0,1)$

$\therefore x\in (0,1)$.