= Solution
Using the representation $X=\mu+\sigma Z$ and <differentiation under the integral sign>,
$$
\frac{\partial f}{\partial\mu}=\mathbb E[U'(X)]>0
$$
because $U$ is strictly increasing. Similarly, part (a) gives
$$
\frac{\partial f}{\partial\sigma}
=\mathbb E[ZU'(X)]
=\sigma\mathbb E[U''(X)]\leq0
$$
because a twice differentiable <concave function> has $U''\leq0$. Therefore
$$
\boxed{f_\mu>0,\qquad f_\sigma\leq0}.
$$
In <mean-variance optimization>, an expected-utility investor consequently prefers a larger mean at fixed standard deviation and a smaller variance at fixed mean. This is the usual risk-averse ordering of Gaussian payoffs.
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