Correct answer: (A) Continuous at the origin; (B) Both partial derivatives at the origin exist and equal zero; (D) Differentiable
Explanation
Let r=√(x²+y²). The numerator has absolute value at most 2r³ and the denominator at least r, so |f|≤2r². Hence f is continuous and f/r→0, proving differentiability with zero derivative, including both zero partials. Away from zero the formula is smooth. **A,B,D**.
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