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Using the representation $X=\mu +\sigma Z$ and differentiation under the integral sign,because $U$ is strictly increasing. Similarly, part (a) givesbecause a twice differentiable concave function has $U^{\prime \prime }\le 0$. ThereforeIn 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.