section_path: Properties/Exponential-like functions
section_name: Exponential-like functions
paragraph_text: Exponential-like functions
  
  Steiner's problem asks to find the global maximum for the function
  
  f (x) = x 1 x. {\displaystyle f(x)=x^{\frac {1}{x}}.}
  
  This maximum occurs precisely at x = e. (One can check that the derivative of ln f (x) is zero only for this value of x.)
  
  Similarly, x = 1/ e is where the global minimum occurs for the function
  
  f (x) = x x. {\displaystyle f(x)=x^{x}.}
  
  The infinite tetration
  
  converges if and only if x ∈ ≈, shown by a theorem of Leonhard Euler.