question_answer

The set of values of 'x' for which function \[f(x)=x\ln x-x+1\] takes positive value only, is

A) \[(1,\infty )\]                    

B) \[\left( \frac{1}{e},\infty  \right)\]

C) \[[e,\infty )\]                    

D) \[(0,1)\cup (1,\infty )\]

Correct Answer: D

Solution :

\[f(x)=x\ln x-x+1;f(1)=0\] \[f'(x)=1+\ln x-1=\ln x\]\[f'(x)=\] \[\Rightarrow \]f (x) is decreasing for (0,1) and increasing for \[(1,\infty )\]