question_answer

Consider the following statements:

1. \[f(x)=\] ln x is an increasing function on \[\left( 0,\infty  \right).\]

2. \[f(x)={{e}^{x}}-x(ln\,\,x)\] is an increasing function on \[\left( 1,\,\infty  \right)\].

Which of the above statements is/are correct?

A) 1 only

B) 2 only

C) Both 1 and 2

D) Neither 1 nor 2

Correct Answer: C

Solution :

[c] \[f(x)=\log \,x\] Clearly \[f(x)\]  is increasing on \[(0,\infty )\] \[f(x)={{e}^{x}}-x\log x\] \[f'(x)={{e}^{x}}-(\log \,x+1)\] From the figure it is clear that \[f'(x)>0\] on \[(1,\infty )\]. So both statements (1) & (2) are correct.