The set-up: Let $G=(\,V,\,E\,)$ be a connected graph. Associated with every edge $e\in E$ is a cost/weight function $f_e(t) = a_e t^2 + b_e t + c_e $, where $a_e>0$. For a fixed $t$ we can define and find a Minimum Spanning Tree on $G$ in the usual way. Let $c_G(t)$ be the cost that MST, and then letting $t$ range over $\mathbb{R} $ we see $c_G(t)$ gives the cost of a MST on $G$ as a function of the parameter $t$.
The problem: Design an (greedy) algorithm to find $t^* = \underset{t\in \mathbb{R}}{\operatorname{argmin}} c_G(t)$.
Note: This question (a more wordy set-up) is in the "Greedy Algorithms" chapter of the K&T Algorithm Design book I've been reviewing, so there should be a greedy algorithm to solve this...
My initial thoughts/attempt: For an arbitrary spanning tree $T$ on $G$ we can find the $t$ which minimizes the cost of that spanning tree (basic calculus) and thus also find the minimum possible cost of that spanning tree. Then perhaps there'd be a way to build up or iterate through these trees, at each step moving towards/to spanning trees with lower minimum possible cost. Couldn't come up with a greedy and efficient algorithm to do this, however.