Past exam of the mathematical methods II course of the University of Oxford / 2013 / 4 / d / Solution

A stationary direction at the point has zero directional derivative, hence is perpendicular to $(1,2)$. The two opposite unit directions areThey are tangents to the circular level curve. “Stationary” here means zero first-order change. A finite straight move in a tangent direction gives squared radius $5+s^2$ and therefore a positive second-order change $\ln (1+s^2/5)$; following the curved level set instead keeps $T$ exactly constant.