First slide

Monotonicity

Question

$If f (x) = 2 x + \cot^{- 1} x + \log (\sqrt{1 + x^{2}} - x), then f (x)$

Moderate

A

$increases in [0, π)$
B

$decreases in [0, π)$
C

$neither increases nor decreases in (0, π)$
D

4) None of these

Solution

$f^{'} (x) = 2 - \frac{1}{1 + x^{2}} + \frac{\frac{x}{\sqrt{1 + x^{2}}} - 1}{\sqrt{1 + x^{2}} - x}$
$= 2 - \frac{1}{1 + x^{2}} - \frac{1}{\sqrt{1 + x^{2}}} > 0$
$In [0, π]$

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