Is there a way to find an analytical expression for the minimum of the following nasty function on $[0, \infty)$?
\begin{align*} Q(u) & = \frac{ (u^2 + t_{11} u + t_{11}p_{11}) (u^2 + t_{22} u + t_{22}p_{22} ) (u^2 + u (t_{12} + p_{12}) + t_{12}^2)^4 }{ (u^2 + u (t_{11} + p_{11}) + t_{11}^2)^2 (u^2 + u (t_{22} + p_{22}) + t_{22}^2)^2 (u^2 + t_{12} u + t_{12}p_{12})^2 }, \end{align*}
where $t_{ij} > 0$, $p_{ij} > 0$, $t_{ij} > p_{ij}$ $i, j = 1, 2.$
Or to say for which parameters there is none?