Question

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

• A. Increases is (–¥, ¥)
• B. Decreases in (0, ¥)
• C. Neither increases nor decreases in (0, ¥)
• D. Some time increases some time decreases

Answer 1

Answer: (A)

Increases is (–¥, ¥)

Answer 2

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

f '(x) = 2 – $\frac{1}{1 + x^{2}}$ + $\frac{1}{\sqrt{1 + x^{2}} - x} \times \frac{2x}{2\sqrt{1 + x^{2}}} - 1$

= 2 – $\frac{1}{1 + x^{2}} + \frac{1}{\sqrt{1 + x^{2}} - x} \times \frac{x - \sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$

= 2 – $\frac{1}{1 + x^{2}} - \frac{1}{\sqrt{1 + x^{2}}}$

Q $\frac{1}{1 + x^{2}} \leq 1$ & $\frac{1}{\sqrt{1 + x^{2}}} \leq 1$

\ 2 –$\frac{1}{1 + x^{2}} - \frac{1}{\sqrt{1 + x^{2}}} > 0$

\ f '(x) > 0

\ x Î R