### Application Of Derivatives Question 1

#### Question 1 - 25 January - Shift 1

Let $f:(0,1) \to \mathbb{R}$ be a function defined by

$f(x)=\frac{1}{1-e^{-x}}$, and

$g(x)=(f(-x)-f(x))$. Consider two statements

(I) $g$ is an increasing function in $(0,1)$

(II) $g$ is one-one in $(0,1)$

Then,

(1) Only (I) is true

(2) Only (II) is true

(3) Neither (I) nor (II) is true

(4) Both (I) and (II) are true

## Show Answer

#### Answer: (4)

#### Solution:

#### Formula: Increasing and decreasing of a function, One-to-One function (3.7)

$g(x)=f(-x)-f(x)=\frac{1+e^{x}}{1-e^{x}}$

$\Rightarrow g^{\prime}(x)=\frac{2 e^{x}}{(1-e^{x})^{2}}>0$

$\Rightarrow g$ is increasing in $(0,1)$

$\Rightarrow g$ is one-one in $(0,1)$