# Showing concavity of a function defined in terms of concave utility function

Suppose $U : \mathbb{R} \to \mathbb{R}$ is concave, and that the random variable $\epsilon$ has zero mean. Assuming that the function $\phi : \mathbb{R} \to \mathbb{R}$, defined by $\phi(\lambda) = \mathbb{E} U(\mu + \lambda \epsilon)$ is everywhere finite-valued, prove that $\phi$ is concave.