The efficient frontier is the portfolios with the minimum of variance ($V$) at a given mean ($E$) or a maximum of mean at a given variance，Why do the optimal portfolios in the effcient frontier, is the efficient frontier consistent with the maximum of expected utility?
For example, assume a logarithmic utility function U=log(1+R), through Taylor series expansion and keep the first three terms, we can get the expected utility $EU=E-(E^2+V)/2$, from this equation, take the partial derivative of $E$, it is not monotonical increase, it is the same with $V$, it seems the efficient frontier does not lead to the maximum of expected utility.
I understand if the utility function is expoential, the expected utility $EU=E-\lambda V$, which is exactly consistent with the efficient frontiers. I just do not know the other form of utility function, like the logarithmic form as above.
Any help would be appreciated.