The normal radius of the trachea does not change so you can view R as a constant as well.
Find v ' and solve v ' = 0.
v ' = k(R-r)(2r) + k(-1)(r^2)
v ' = 2rk(R-r) + -kr^2
v ' = 2rkR - 2kr^2 - kr^2
v ' = 2rkR - 3kr^2
Set v ' = 0 and solve for r.
0 = 2rkR - 3kr^2
0 = rk(2R - 3r)
rk = 0 or 2R - 3r = 0
r = 0 or 2R = 3r
r = 0 or r = 2R/3
Plug 0 and 2R/3 for the orginal v and the larger value is the maximum.
If r = 0, then v = k(R - 0)(0^2) = 0
If r = 2R/3, then v = k(R - 2R/3)(2R/3)^2
v = k(R/3)(4R^2 / 9)
v = 4kR^3 / 27
Therefore, the radius of 2R/3 will produce the maximum air velocity of 4kR^3 / 27.