If both the roots of the equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$ lie inside the randomly chosen real points ${\mathrm{k}}_1, {\mathrm{k}}_2$ then which of the following facts are the most appropriate and valid ones?

(i) $\mathrm{af}\left({\mathrm{k}}_1\right)>0, af\left({\mathrm{k}}_2\right)>0$

(ii) $\mathrm{f}\left({\mathrm{k}}_1\right).\mathrm{f} \left({\mathrm{k}}_2\right)<0$

(iii) $\mathrm{af}\left({\mathrm{k}}_1\right)<0, \mathrm{af} \left({\mathrm{k}}_2\right)<0$

(iv) $\mathrm{D}\ge 0$

(v) $\mathrm{D}>0$

(vi) ${\mathrm{k}}_1<-\frac{\mathrm{b}}{2\mathrm{a}}<{\mathrm{k}}_2$

(vii) $\mathrm{f}\left({\mathrm{k}}_1\right).\mathrm{f}\left({\mathrm{k}}_2\right)>0$