Constrained order-parameter boundary stability

A stationary point on the domain $\varphi \ge 0$ must be tested using allowed one-sided variations. For $F=A{\varphi }^2/2-B{\varphi }^3+\lambda {\varphi }^4/4$ with $B>0$, the boundary is a local minimum when $A>0$ and a local maximum when $A<0$. At $A=0$, the leading negative cubic term makes it a one-sided local maximum even though $F^{\prime \prime }(0)=0$; on an unrestricted real line it is instead a stationary inflection point.