Consider $A: \mathbb{R}_{\geq 0}^n \rightarrow \mathbb{R}_{\geq 0}^{n \times n} $ and $B \in \mathbb{R}_{\geq 0}^{n \times m}$, and $c \in \mathbb{R}_{\geq 0}^n$.
Assume that, for all $y \in \mathbb{R}_{\geq 0}^n$, the function $x \mapsto c^\top A(x) \, y$ is convex.
I am wondering if the function $x \mapsto c^\top A(x) B x$ is quasiconvex. If not, find under what additional condition would it be quasiconvex.
Comment: I tried to apply the fact that the composition of a quasiconvex function and a non-decreasing one preserves quasiconvexity, but the order seems reversed.