Question

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

Answer 1

$$S(r) = c(R^2 − r^2), 0\leq r\leq R$$

Find the first derivative with respect to $r$

$$S'(r)=(c(R^2 − r^2))'=-2cr$$

Find the critical number(s)

$$S'(r)=0=>-2cr=0=>r=0$$

$$S(0)=c(R^2-0)=cR^2>0$$

$$S(R)=c(R^2-R^2)=0$$

The speed of the blood is absolute maximum with value of $cR^2$ when $r=0$

(on the central axis).