Question 3

f(x) = 2 − 2x² − x² sin(1/x) for x ≠ 0, f(0) = 2. Differentiability at 0: [f(x) − f(0)]/x = −2x − x sin(1/x) → 0, so f is differentiable at 0 with f'(0) = 0. So (A) is false. For x ≠ 0: f(x) − 2 = −x²(2 + sin(1/x)) ≤ −x² < 0, so f(x) < f(0) for all x ≠ 0. Hence x = 0 is a strict local maximum, not minimum. (D) false. Derivative for x ≠ 0: f'(x) = −4x − 2x sin(1/x) + cos(1/x). Near 0, the cos(1/x) term oscillates between −1 and 1 while the other terms tend to 0, so f' changes sign infinitely often in any (0, δ) and any (−δ, 0). Thus f is neither monotonic decreasing on any (0, δ) nor increasing on any (−δ, 0). So (B) is false, and (C) is true.

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