Question

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

• A. Decreases on $( - \infty,\infty)$
• B. Decreases on $\lbrack 0,\infty)$
• C. Neither decreases nor increases on $\lbrack 0,0)$
• D. Increases on $( - \infty,\infty)$

Answer 1

Answer: (D)

Increases on $( - \infty,\infty)$

Answer 2

$y'(x) = 2 - \frac{1}{1 + x^{2}} + \frac{1}{\sqrt{1 + x^{2}} - x}\frac{d}{dx}\left( \sqrt{1 + x^{2}} - x \right)$

=$2 - \frac{1}{1 + x^{2}} + \frac{1}{\sqrt{1 + x^{2}} - x}\text{ x }\left( \frac{x}{\sqrt{1 + x^{2}}} - 1 \right)$

=$2 - \frac{1}{1 + x^{2}} - \frac{1}{\sqrt{1 + x^{2}}} = \frac{\left( 1 + x^{2} - \sqrt{1 + x^{2}} \right) + x^{2}}{\sqrt{1 + x^{2}}} \geq 0$

for all x. so $f(x)$ increases on $( - \infty,\infty)$