Let $x\in [0,1]$ and $u_1,u_2,u_3$ be three positive integers.
Is it possible to know what triple $(i,j,k)\in\{1,2,3\}^3$ with $i,j$ and $k$ distinct, maximizes the quantity $u_i x(1-x)(2-x)+u_j x(1+(1-x)^2)+u_k x(2-x)$ (depending on $u_1,u_2,u_3,x$ which are fixed)?
I know that for all $x\in[0,1]$, $x(1-x)(2-x)\leq x(1+(1-x)^2) \leq x(2-x)$, but I dont know where to go from there.