JEE Mains · Maths · STD 12 - 6. Application of derivatives
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
- D Both \((I)\) and \((II)\) are true
Answer & Solution
Correct Answer
(D) Both \((I)\) and \((II)\) are true
Step-by-step Solution
Detailed explanation
\(g ( x )= f (- x )- f ( x )=\frac{1+ e ^{ x }}{1- e ^{ x }}\) \(\Rightarrow g ^{\prime}( x )=\frac{2 e ^{ x }}{\left(1- e ^{ x }\right)^2} > 0\) \(\Rightarrow g \text { is increasing in }(0,1)\) \(\Rightarrow g \text { is one-one in }(0,1)\)
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