A blood vessel with a circular cross section of constant radius R carries blood that flows parallel to the axis of the vessel with a velocity of v(r) = V(1 - r²/R²), where V is a constant and r is the distance from the axis of the vessel. 

a. Where is the velocity a maximum? A minimum?

b. Find the average velocity of the blood over a cross  section of the vessel.

c. Suppose the velocity in the vessel is given by v(r) = V(1 - r²/R²)^1/p, where p ≥ 1. Graph the velocity profiles for p = 1, 2, and 6 on the interval 0 ≤ r ≤ R. Find the average velocity in the vessel as a function of p. How does the average velocity behave as p→∞?