Steiner's problem, asked and answered by , is the problem of finding the

maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...

of the

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

f(x)=x^.\,

It is named after

Jakob Steiner Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards s ...

. The maximum is at

x = e

, where ''e'' denotes the base of the natural logarithm. One can determine that by solving the equivalent problem of maximizing :

g(x) = \ln f(x) = \frac.

Applying the

first derivative test In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information a ...

, the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

g

is :

g'(x) = \frac,

g'(x)

is positive for

0 and negative for x>e, which implies that g(x) – and therefore f(x) – is

increasing In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...

for

0 and decreasing for x>e. Thus, x=e is the unique

global maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...

f(x).

References

* {{DEFAULTSORT:Steiner's calculus Problem Functions and mappings Mathematical optimization