First slide

Monotonicity

Question

$If y = 2 x - \tan^{- 1} x - \log \{x + \sqrt{(1 + x^{2})}\}, then:$

Moderate

A

$y increases in [0, \infty)$
B

$y decreases in [0, \infty)$
C

$y increases in (- \infty, 0)$
D

$y increases in (- \infty, \infty)$

Solution

$y = 2 x - \tan^{- 1} x - \log \{x + \sqrt{1 + x^{2}}\}$
$\Rightarrow \frac{d y}{d x} = 2 - \frac{1}{1 + x^{2}} - \frac{1 + \frac{2 x}{2 \sqrt{1 + x^{2}}}}{x + \sqrt{1 + x^{2}}}$
$= 2 - \frac{1}{1 + x^{2}} - \frac{1}{\sqrt{1 + x^{2}}}$
$= \frac{2 + 2 x^{2} - 1 - \sqrt{1 + x^{2}}}{1 + x^{2}}$
$= \frac{1 + 2 x^{2} - \sqrt{1 + x^{2}}}{1 + x^{2}} \geq 0 \forall x \in R$
$Hence y increases in (- \infty, \infty)$

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