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

The Correct Option is A, C, D

Certainly! Let's go through the solution step-by-step. 

To solve this problem, we need to analyze the differentiability and continuity of the derivative of the function \( f \) which is defined on the interval \((-1, 1)\). The given condition is that \( f(0) = 0 \), and there exists a constant \( M > 0 \) such that \( |f'(x)| \leq M|x| \) for all \( x \in (-1, 1) \). We'll explore each of the provided options below:

1. Check if \( f' \) is continuous at \( x = 0 \):

The given condition \( |f'(x)| \leq M|x| \) suggests that as \( x \to 0 \), \( f'(x) \to 0 \) because \( |f'(x)| \) is dominated by \( |x| \) which tends to zero. Therefore, \( f' \) is continuous at \( x = 0 \).

1. Check if \( f' \) is differentiable at \( x = 0 \):

The continuity of \( f' \) at \( x = 0 \) implies that \( f' \) approaches a unique limit as \( x \to 0 \). However, differentiability of \( f' \) at \( x = 0 \) requires that the derivative of \( f' \) exists at \( x = 0 \), which is not guaranteed by the given condition. Hence, there is not enough information to conclude that \( f' \) is differentiable at \( x = 0 \).

1. Check if \( ff' \) is differentiable at \( x = 0 \):

To check differentiability, examine the derivative \((ff')'(x) = f'(x)f'(x) + f(x)f''(x)\). At \( x = 0 \), since \( f(0) = 0 \) and \( f'(0) = 0 \), this implies \((ff')'(0)\) exists because the terms vanish. Hence, \( ff' \) is differentiable at \( x = 0 \).

1. Check if \((f')^2\) is differentiable at \( x = 0 \):

Consider the derivative of \((f'(x))^2\), which is \( 2f'(x)f''(x) \). As \( f'(0) = 0 \) by continuity, the expression at \( x = 0 \) evaluates to \( 0 \), indicating that \((f')^2\) is differentiable at \( x = 0 \).

Conclusion: Based on the above analysis, the correct answer is that \( f' \) is continuous at \( x = 0 \), \( ff' \) is differentiable at \( x = 0 \), and \((f')^2\) is differentiable at \( x = 0 \).