# 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 .

## Proof

 $\ds \map {f'} x$ $=$ $\ds \frac \d {\d x} x^{1/x}$ $\ds$ $=$ $\ds \frac \d {\d x} e^{\ln x / x}$ $\ds$ $=$ $\ds e^{\ln x / x} \paren {\frac 1 {x^2} - \frac {\ln x} {x^2} }$ $\ds$ $=$ $\ds \frac {x^{1/x} } {x^2} \paren {1 - \ln x}$

$\dfrac {x^{1/x} } {x^2}$ 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 $x = e$.

$\blacksquare$

## Source of Name

This entry was named for Jakob Steiner.