Question

Let f:(0, ∞) → ℝ be a differentiable function such that f' (x) = 2 \(-\frac{f(x)}{x}\) for all x ∈ (0, ∞) and f(1) ≠ 1. Then

(A) \(\lim\limits_{x \to 0+}f'(\frac{1}{x})= 1\)

(B) \(\lim\limits_{x \to 0+}xf(\frac{1}{x})= 2\)

(C) \(\lim\limits_{x \to 0+}x^2 f'(x)= 0\)

(D) |f(x)| ≤ 2 for all x ∈(0, 2)

Answer 1

 (A) \(\lim\limits_{x \to 0+}f'(\frac{1}{x})= 1\)

(D) for c ≠ 0 f(x) is unbounded function for x ∈ (0, 2)