Abstract The potential of exploiting the close relationship between Hurwitz polynomials and positive definite quadratic forms is discussed in a new direction, i.e. the examination of the existence of an extremum at a stationary point of a constrained optimization problem by subjecting the pertinent test quadratic form q to an inquiry of which are the polynomials G whose Hurwitzness would be implied by the definiteness of q (and vice versa). Particularly, in network and system theory, some Hurwitz polynomial H may already be involved at the outset of the optimization problem, and the Hurwitzness of H may imply the Hurwitzness of G which, in turn, would imply the definiteness of q. Our approach is illustrated by an examination of the optimization of several functionals associated with the difference-type decomposition of polynomials encountered in sensitivity considerations of Linvill's classical RC active filters. A common feature of these functionals, which facilitates our task, is that the matrices of the pertinent quadratic forms q are of the finite Hankel form which enables us, firstly, to identify the required polynomials G by their Markov parameters and then to establish, by a very simple test, their Hurwitzness or non-Hurwitzness.