I want to proof that the function $\phi:\mathbb R\rightarrow\mathbb R: \phi(\lambda)=\mathbb{E}U(w+\lambda X)$, which is everywhere finite-valued, is concave.
$U:\mathbb R\rightarrow\mathbb R$ is concave and the random variable $X$ has zero mean.
My first idea was to use Taylor expansion to show that the second derivative is smaller than $0$. When I use Taylor expansion I get approximately $U(w)+U'(w)\mathbb{E}(\lambda X)+U''(w)\mathbb{E}\lambda^2X^2=U(w)+\lambda^2U''(w)\mathbb E{X^2}$ and I now that $U''(w)\le0$ because $U$ is concave, but how can I continue?
Hint: $U$ is concave iff $$ U(\eta w_1+(1-\eta) w_2) \ge \eta U(w_1)+(1-\eta)U(w_2),\;\;\forall w_1, w_2 \mbox{ and } \eta \in [0,1] $$ Now $$ EU(\eta w_1+(1-\eta) w_2 + \lambda X)=EU\big(\eta (w_1+ \lambda X)+(1-\eta) (w_2 + \lambda X)\big)\ge\ldots $$