Question

Let $f:(0,1) \rightarrow R$ 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,

• A. Only (I) is true
• B. Both (I) and (II) are true
• C. Neither (I) nor (II) is true
• D. Only (II) is true

Answer 1

Answer: (B)

Both (I) and (II) are true

Answer 2

g(x)=f(−x)−f(x)=1−ex1+ex​
⇒g′(x)=(1−ex)22ex​>0
⇒g is increasing in (0,1)
⇒g is one-one in (0,1)