Steiner's calculus problem

Steiner's problem, asked and answered by Steiner (1850), is the problem of finding the maximum of the function

It is named after Jakob Steiner.

The maximum is at x = e , where e denotes the base of natural logarithms. One can determine that by solving the equivalent problem of maximizing

The derivative of g can be calculated to be

It follows that g ′ ( x ) is positive for 0 < x < e and negative for x > e , which implies that g ( x ) (and therefore f ( x ) ) increases for 0 < x < e and decreases for x > e . Thus, x = e is the unique global maximum of f ( x ) .