Let $f(s)=(1-\alpha(s))*u(I-s)+\alpha(s)*u(I-L-s)$ where $I>0$, $L>0$ and $I>L$
Given that $s\ge0$, I want to construct the functions $\alpha(s)$ and $u(x)$ such that: $0\le\alpha(s)\le1$, $\alpha'(s)<0$ and $\alpha''(s)>0$, $u(x)\ge0$, $u'(x)>0$ and $u''(x)<0$, $f''(s)<0$ and the solution to the maximization of $f$ with respect to $s$ must not be more complicated than solving a quadratic equation in $s$.
I don't know how to start with this problem. My professor said not to use logarithm or square root function for $u$. I tried $\alpha(s)=\frac{1}{s+1}$ and $u(x)=a-\frac{1}{x}$ where $a$ is a parameter which keeps $u$ positive. But I did not get a quadratic equation. Any tips on how to go about this problem?