QJEEM23Ma22ee: JEE Main 2023 Mathematics - Class 12 JEEM23Ma22ee Solution: Let f : ( 0 , 1 ) → R be a function defined f ( x ) = 1 1 − e − …

Question

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Let $f : (0, 1) \to R$ be a function defined $f (x) = \frac{1}{1 - e^{- x}}$ , 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,

Accepted Answer

g(x)=f(-x)-f(x)

\begin{aligned} = \frac{1}{1 - e^{x}} - \frac{1}{1 - e^{- x}} \\ = \frac{1}{1 - e^{x}} - \frac{e^{x}}{e^{x} - 1} \\ = \frac{1 + e^{x}}{1 - e^{x}} \\ g^{'} (x) & = \frac{(1 - e^{x}) e^{x} - (1 + e^{x}) (- e^{x})}{{(1 - e^{x})}^{2}} \\ = \frac{e^{x} - 2 e^{x} + e^{x} + 2 e^{x}}{{(1 - e^{x})}^{2}} > 0 \end{aligned}

So both statements are correct

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