JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(f(x)=2 x+\tan ^{-1} x\) and \(g(x)=\log _e\left(\sqrt{1+x^2}+x\right)\), \(x \in[0,3]\). Then
- A There exists \(\hat{ x } \in[0,3]\) such that \(f ^{\prime}(\hat{ x }) < g ^{\prime}(\hat{ x })\)
- B \(\max f(x) > \max g(x)\)
- C There exist \(0 < x_1 < x_2 < 3\) such that \(f(x) < g(x)\), \(\forall x \in\left( x _1, x _2\right)\)
- D \(\min f ^{\prime}( x )=1+\max g ^{\prime}( x )\)
Answer & Solution
Correct Answer
(B) \(\max f(x) > \max g(x)\)
Step-by-step Solution
Detailed explanation
\(f ( x )=2 x +\tan ^{-1} x \text { and } g ( x )=\ln \left(\sqrt{1+x^2}+x\right)\) \(\text { and } x \in[0,3]\) \(g ^{\prime}( x )=\frac{1}{\sqrt{1+x^2}}\) Now, \(0 \leq x \leq 3\) \(0 \leq x^2 \leq 9\) \(1 \leq 1+x^2 \leq 10\) So,…
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