Let $X,Y$ be two independent random normal standard distributions. Consider a function $u(x)=\sqrt[ ]{x}$ if $x\geq{}0$ and $u(x)=-2\sqrt[ ]{-x}$ if $x<0$. Define $Z=aX+(1-a)Y$.

**Question:** How do I find the necessary and sufficient conditions such that $a=\displaystyle\frac{1}{2}$ is a global maximum of $Eu(Z)$.