Question

Let f(x) = 2x – tan⁻¹x − ln(x+ $\sqrt{1 + x^{2}}$) ∀ x ∈ R. Then

• A. f(x) in non-increasing in (-∞, ∞)
• B. f(x) in non-decreasing in (-∞, ∞)
• C. f(x) is increasing in (-∞, ∞)
• D. f(x) is decreasing in (-∞, ∞)

Answer 1

Answer: (C)

f(x) is increasing in (-∞, ∞)

Answer 2

f'(x) =

⇒ f '(x) = $\frac { x \left( 2 + \sqrt { 1 + x ^ { 2 } } \right) } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }$

clearly f"(x) < 0 ∀ x ∈ R⁻ and f ''(x) > 0 ∀ x > 0.

f '(0) = 0 ⇒ f '(x) ∀ x > 0 and f '(x) > 0 ∀ x < 0.

Thus f(x) is increasing for ∀ x ∈ R .