Hi all, I have been working on the old "what is the largest cylinder by volume that can inscribed in a sphere of radius r" question. The solution is easy enough, but proving that the point at which dv/dr = 0 (where v is the volume of the cylinder and r = the radius of the circular base of the cylinder) is a maximum by using the second derivative test seems ridiculously laborious when it is obviously a maximum. Logically we can say that this point can't be a minimum because the minimum would have to be r = 0, v =0. But how I can prove that it's not a horizontal POI without reverting to the 2nd derivative test.

is there a way to prove this logically withouth having to find the second derivative?