Question

Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(2) = 2 \) and \[ |f(x) - f(y)| \leq 5| |x - y|^{3/2}. \] For all \( x \in \mathbb{R}, y \in \mathbb{R} \), let \( g(x) = x^3 f(x) \). Then \[ g'(2) = \]

• A. 5
• B. \( \frac{15}{2} \)
• C. 12
• D. 24

Hint:

When differentiating products of functions, always use the product rule and substitute known values to compute the result.

Answer 1

The Correct Option is D

Step 1: Understanding the function \( g(x) \).
We are given that \( g(x) = x^3 f(x) \), and we need to compute \( g'(2) \). Using the product rule for differentiation: \[ g'(x) = 3x^2 f(x) + x^3 f'(x). \]
Step 2: Substituting \( x = 2 \).
Substitute \( x = 2 \) into the expression for \( g'(x) \): \[ g'(2) = 3(2)^2 f(2) + (2)^3 f'(2) = 3 \times 4 \times 2 + 8 f'(2). \] Since \( f'(2) = 2 \), we get: \[ g'(2) = 24 + 8 \times 2 = 24. \]
Step 3: Conclusion.
Thus, the correct answer is (D).