Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 1 / d / i / Solution

With ${\varphi }_o=0$,For $C<a$, both branches are locally convex at the origin. Atthe negative-side curvature vanishes; for $C>a$ the interval near $0^{-}$ has $f^{\prime \prime }=a-C+3b{\varphi }^2<0$, so a homogeneous composition there is unstable and the equilibrium free energy is its convex envelope. The sketch therefore has an ordinary upward quartic on the positive side and, beyond $C_c$, a negative-curvature shoulder and a minimum on the negative side.