Solution
= Solution
Write $X=\mu+\sigma Z$, where $Z$ has the <standard normal distribution>. Apply <Gaussian integration by parts> to $g(z)=U'(\mu+\sigma z)$:
$$
\mathbb E[ZU'(X)]=\mathbb E[g'(Z)]=\sigma\mathbb E[U''(X)].
$$
Since $X-\mu=\sigma Z$,
$$
\boxed{\mathbb E[U'(X)(X-\mu)]=\sigma^2\mathbb E[U''(X)]}.
$$
The polynomial growth assumption and decay of the <normal distribution> density justify the integration by parts.