Question

Poiseuille’s law asserts that the speed of blood that is r centimeters from the central axis of an artery of radius S(r)=c(R^2-r^2), where c is a positive constant. Where is the speed of the blood greatest?

Answer 1

We consider the function $S(r)=c\cdot(R^2-r^2)$ on the segment $r\in[0,R]$ and find its derivative $S'(r)=(c\cdot(R^2-r^2))_r^{'}=-2c\cdot r\le0,t\in[0,R]$ and $S'(r)<0.r\in (0,R]$ . Therefore the function S(r) strictly decreases in [0,R] and we have inequality $S(0)>S(r) \forall(r\in(0.R]$ . Therefore $S(0)=c\cdot(R^2-0^2)=c\cdot R^2=max_{r\in[0,R]}S(r)$ .

So speed of the blood has its maximum as r=0 or in the central axis of an artery.