In order to define the problem please observe this two-dimensional diagram…

If we indicate with the radius of base and with the height of inscribed cone is…

The volume of cylinder is therefore…

Taking the derivative respect to we obtain with little amount of work…

(1)

Now we have to find the value of for which the derivative vanish and chose the value which maximises the volume of cone. The (1) vanish for…

… and that happens for . The maximum volume of an inscribed cylinder is then…

Very easy…

Kind regards