Assume $A \in C^{\infty}(\mathbb{R}^n, \text{Mat}_N(\mathbb{R}))$, s.t. there exists a $C$, s.t.
$$\Vert A (x)\Vert \leq C, \ \ \text{ for all }x.$$
Can we say now that $\vert x^T A x \vert = \Vert x^T \Vert \Vert Ax \Vert \leq C \Vert x \Vert^2 $?
I think it can't be right, because there are different norm one could use as matrix norms.