Question

Let f : (−1, 1) → $\R$ be a differentiable function satisfying f(0) = 0. Suppose there exists an M > 0 such that |f' (x)| ≤ M|x| for all x ∈ (−1, 1). Then

• A. f' is continuous at x = 0
• B. f' is differentiable at x = 0
• C. ff′ is differentiable at x = 0
• D. (f′)2 is differentiable at x = 0

Answer 1

Answer: (A), (C), (D)

f' is continuous at x = 0

Answer 2

The correct option is (A) : f' is continuous at x = 0, (C) : ff′ is differentiable at x = 0 and (D) : (f′)2 is differentiable at x = 0.