JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(f : R \rightarrow R\) and \(g : R \rightarrow R\) be two functions defined by \(f(x)=\log _{e}\left(x^{2}+1\right)-e^{-x}+1\) and \(g(x)=\frac{1-2 e^{2 x}}{e^{x}}\). Then, for which of the following range of \(\alpha\), the inequality \(f\left(g\left(\frac{(\alpha-1)^{2}}{3}\right)\right)>f\left(g\left(\alpha-\frac{5}{3}\right)\right)\) holds?
- A \((2,3)\)
- B \((-2,-1)\)
- C \((1,2)\)
- D \((-1,1)\)
Answer & Solution
Correct Answer
(A) \((2,3)\)
Step-by-step Solution
Detailed explanation
\(f ( x )=\log _{ e }\left( x ^{2}+1\right)- e ^{- x }+1\) \(\Rightarrow f ^{\prime}( x )=\frac{2 x }{ x ^{2}+1}+ e ^{- x }>0 \quad \forall x \in R\) \(\Rightarrow f\) is strictly increasing \(g ( x )=\frac{1-2 e ^{2 x }}{ e ^{ x }}= e ^{- x }-2 e ^{ x }\)…
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