Question

Consider the following statements :
1. $f\left(x\right)=lnx$ is an increasing function on $(0,\infty )$.
2. $f\left(x\right)=e^x-x\left(lnx\right)$ is an increasing function on $(1,\infty )$.
Which of the above statements is\are correct?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is C Both 1 and 2

Solution:

1. We have,

$f\left(x\right)=lnx$

$\therefore f^{\prime }\left(x\right)=\frac{1}{x}$

Here, $f^{\prime }\left(x\right)>0$ in the interval $(0,\infty )$

$\therefore f\left(x\right)$ is increasing function in $(0,\infty )$

2. We have,

$f\left(x\right)=e^x-x\left(lnx\right)$

$\therefore f^{\prime }\left(x\right)=e^x-1-lnx$

Here, $f^{\prime }\left(x\right)>0$ in the interval $(1,\infty )$

$\therefore f\left(x\right)$ is increasing function in $(1,\infty )$

Hence, C is the correct option.