pr.probability - Maximum of Convex combination of random variables

First, $Z$ is distributed like $bX$, with $b>0$ and $b^2=a^2+(1−a)^2$. Second, for every $x$, $u(bx)=sqrt{b}u(x)$. Third, $E(u(X))=-E(sqrt{X};X>0)$ is negative. Hence, $E(u(Z))=sqrt{b}E(u(X))$ is at its maximum when $b$ is at its minimum. This happens when $a=frac12$.