Expected-utility-consistent mean-variance preference functions with perverse derivatives

It is well-known that if every attainable gamble x can be completely characterized by its mean μ and variance σ², then expected utility is given by a mean-variance preference function: EU (x) = V (μ, σ²). It is shown here thatV _σ ² can be positive even if U″<0 and thatV _μ can be negative even if U′>0. Necessary and sufficient conditions on the family of attainable gambles are given to rule out these possibilities.