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This work is concerned with the study of the extreme rays of the convex coneof $3\times 3$ quasiconvex quadratic forms (denoted by ${\cal C}_3$). Wecharacterize quadratic forms $f\in {\cal C}_3,$ the determinant of the acoustictensor of which is an extremal polynomial, and conjecture/discuss about othercases. We prove that in the case when the determinant of the acoustic tensor ofa form $f\in {\cal C}_3$ is an extremal polynomial other than a perfect square,then the form must itself be an extreme ray of ${\cal C}_3;$ when thedeterminant is a perfect square, then the form is either an extreme ray of${\cal C}_3$ or polyconvex; and finally, when the determinant is identicallyzero, then the form $f$ must be polyconvex. The zero determinant case plays animportant role in the proofs of the other two cases. We also make a conjectureon the extreme rays of ${\cal C}_3,$ and discuss about weak and strong etremalsof ${\cal C}_d$ for $d\geq 3.$ where it turns out that several properties of${\cal C}_3$ do not hold for ${\cal C}_d$ for $d>3,$ and thus case $d=3$ isspecial. These results recover all previously known results (to our bestknowledge) on examples of extreme points of ${\cal C}_3$ that were proved to besuch. Our results also improve the ones proven by the first author and Milton[20].

On the extreme rays of the cone of $3\times 3$ quasiconvex quadratic forms: Extremal determinats vs extremal and polyconvex forms