Consider a utility function $U: \mathbb{R} \rightarrow \mathbb{R}$, which is assumed to be concave, strictly increasing and twice differentiable. Further, $U$ satisfies

$$\left|U^{\prime}(x)\right| \leqslant c|x|^{\alpha}, \quad \forall x \in \mathbb{R},$$

for some positive constants $c$ and $\alpha$. Let $X$ be an $\mathcal{N}\left(\mu, \sigma^{2}\right)$-distributed random variable and set $f(\mu, \sigma):=\mathbb{E}[U(X)]$.

(a) Show that

$$\mathbb{E}\left[U^{\prime}(X)(X-\mu)\right]=\sigma^{2} \mathbb{E}\left[U^{\prime \prime}(X)\right]$$

(b) Show that $\frac{\partial f}{\partial \mu}>0$ and $\frac{\partial f}{\partial \sigma} \leqslant 0$. Discuss this result in the context of meanvariance analysis.

(c) Show that $f$ is concave in $\mu$ and $\sigma$, i.e. check that the matrix of second derivatives is negative semi-definite. [You may use without proof the fact that if a $2 \times 2$ matrix has nonpositive diagonal entries and a non-negative determinant, then it is negative semi-definite.]