Check the differentiability of f(x) = x² sin(1/x) for x ≠ 0, f(0) = 0 at x = 0. If differentiable, what is f'(0)?

Check the differentiability of f(x) = x² sin(1/x) for x ≠ 0, f(0) = 0 at x = 0. If differentiable, what is f'(0)?

• A. Not differentiable
• B. Differentiable, f'(0) = 1
• C. Differentiable, f'(0) = 0
• D. Differentiable, f'(0) = −1

Answer: C) Differentiable, f'(0) = 0

Explanation: Using definition: lim(h→0) (f(h) − f(0))/h = lim(h→0) (h² sin(1/h))/h = lim(h→0) h sin(1/h). Since sin(1/h) is bounded between −1 and 1, the limit is 0.