Expected utility of a Gaussian location-scale family

For $X=\mu +\sigma Z$ with $Z$ having the standard normal distribution and an increasing concave utility function $U$, define $f(\mu ,\sigma )=\mathbb{E}[U(X)]$. Then $f_{\mu }=\mathbb{E}[U^{\prime }(X)]>0$ and Gaussian integration by parts gives $f_{\sigma }=\sigma \mathbb{E}[U^{\prime \prime }(X)]\le 0$. Moreoveris negative semidefinite, so $f$ is jointly concave in $\mu$ and $\sigma$.