 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-4/10f/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For fixed

(x_{1}, x_{2})

, apply the given one-variable identity to the smooth function

u (t) = f (x_{1}, t x_{2}) .

u^{'} (0) = x_{2} D_{2} f (x_{1}, 0), u^{''} (t) = x_{2}^{2} D_{2}^{2} f (x_{1}, t x_{2}) .

Therefore

f (x_{1}, x_{2}) = f (x_{1}, 0) + x_{2} D_{2} f (x_{1}, 0) + x_{2}^{2} h (x_{1}, x_{2}),

where

h (x_{1}, x_{2}) = \int_{0}^{1} (1 - t) D_{2}^{2} f (x_{1}, t x_{2}) d t .

Every derivative of the integrand is continuous. Repeated differentiation under the integral sign therefore shows that

h

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