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 $(0,1)$
