Let f :(0,1) R be a function defined by f(x)=1 1-e^ -x , and g(x)… - Vidyadip

MCQ

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 statement:

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

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

A
Only $(I)$ is true
B
Only $(II)$ is true
C
Neither $(I)$ nor $(II)$ is true
✓
Both $(I)$ and $(II)$ are true

✓

Answer

Correct option: D.

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

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

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

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

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

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