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 , where e denotes the base of the natural logarithm. One can determine that by solving the equivalent problem of maximizing
Applying the first derivative test, the derivative of is
so is positive for and negative for , which implies that – and therefore – is increasing for and decreasing for Thus, is the unique global maximum of
References
- Eric W. Weisstein. "Steiner's Problem". MathWorld. Retrieved December 8, 2010.
- Steiner, J. (1850), "Über das größte Product der Theile oder Summanden jeder Zahl" (PDF), Crelle, 40: 208
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