 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ib/paper-3/3b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Put

r = (x^{2} + y^{2})^{1/2}

. Since

x^{2} y^{2} \leq (x^{2} + y^{2})^{2} /4

, the nonzero-point expression satisfies

0 \leq ∣ f (x, y) ∣ \leq \frac{r ^{2}}{4}, \frac{∣ f ( x , y ) - f ( 0 , 0 ) ∣}{r} \leq \frac{r}{4} ⟶ 0.

Thus the defining remainder condition for differentiability holds with the zero linear map:

D f_{(0, 0)} = 0.

The estimate controls every approach direction at once, which is stronger than merely finding the two zero partial derivatives.

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