Steiner's Calculus Problem

Theorem

Let $f: \R_{>0} \to \R$ be the real function defined as:

$\forall x \in \R_{>0}: \map f x = x^{1/x}$

Then $\map f x$ reaches its maximum at $x = e$ where $e$ is Euler's number.

$\ds \map {f'} x$

$=$

$\ds \frac \d {\d x} x^{1/x}$

$\ds $

$=$

$\ds \frac \d {\d x} e^{\ln x / x}$

$\ds $

$=$

$\ds e^{\ln x / x} \paren {\frac 1 {x^2} - \frac {\ln x} {x^2} }$

$\ds $

$=$

$\ds \frac {x^{1/x} } {x^2} \paren {1 - \ln x}$

$\dfrac {x^{1/x} } {x^2}$ is always greater than $0$.

Therefore:

$\map {f'} x > 0$ for $\ln x < 1$

$\map {f'} x = 0$ for $\ln x = 1$

$\map {f'} x < 0$ for $\ln x > 1$

By Derivative at Maximum or Minimum, maximum is obtained when $\ln x = 1$,

that is, when $x = e$.

$\blacksquare$

This entry was named for Jakob Steiner.

1850: Jakob Steiner: Über das größte Product der Theile oder Summanden jeder Zahl (J. reine angew. Math. Vol. 40: p. 208)
1965: Heinrich Dörrie: 100 Great Problems of Elementary Mathematics
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 444 \, 667 \, 861 \ldots$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 44466 \, 7861 \ldots$