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

The denominator is positive throughout the domain. Holding $y$ fixed, differentiate with respect to $x$: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.