1. The problem statement, all variables and given/known data When you cough, your windpipe contracts. The speed, v, with which air comes out depends on the radius, r, of your windpipe. If R is the normal (rest) radius of your windpipe, then for r <_ R, the speed if given by v = a(R-r)r2 where a is a positive constant. What value of r maximizes v? What is the maximum speed? Show all work. Use the procedure for finding global extrema to justify that you have found the global maximum velocity. 2. Relevant equations v = a(R-r)r2 3. The attempt at a solution Derivative is 2ar(R-r) - ar2 (R is considered a constant, right?) = 2arR - 2ar2 -ar2 = 2arR - 3ar2 = 0 2arR = 3ar2 2aR = 3ar r = (2/3)R Not too confident in that answer. For maximum speed I just plugged it into the original equation: v = a((2/3)R)2((1/3)R) Any help is appreciated, whether it be by pointing out a mistake or helping with the last part (justify answer). Edit The interval would be 0 < r <_ R, right?