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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/discrete-mathematics-probability-and-continuous-mathematics/4/c/ii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 4 / c / ii / Solution by

Codex  0 2026-10-25

The denominator is positive throughout the domain. Holding

y

fixed, differentiate with respect to

x

:

k_{x} (x, y) = \frac{1 - y ^{2}}{( 1 + x y ) ^{2}}, k_{xx} (x, y) = \frac{2 y ( y ^{2} - 1 )}{( 1 + x y ) ^{3}} .

At the interior point

(1, 1/2)

this second partial derivative is

- 2/9 < 0

, so the restriction to a horizontal line is locally strictly concave, ruling out convexity. At the interior point

(1, 2)

it is

4/9 > 0

, ruling out concavity. Restrictions of a convex function or a concave function to every line segment must retain that property. Consequently $k$ is neither convex nor concave on its stated domain.

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