section_path: Applications/Optimal planning problems
section_name: Optimal planning problems
paragraph_text: Optimal planning problems
  
  The maximum value of x x {\displaystyle {\sqrt{x}}} occurs at x = e {\displaystyle x=e}. Equivalently, for any value of the base b > 1, it is the case that the maximum value of x − 1 log b x {\displaystyle x^{-1}\log _{b}x} occurs at x = e {\displaystyle x=e} (Steiner's problem, discussed below).
  
  This is useful in the problem of a stick of length L that is broken into n equal parts. The value of n that maximizes the product of the lengths is then either
  
  The quantity x − 1 log b x {\displaystyle x^{-1}\log _{b}x} is also a measure of information gleaned from an event occurring with probability 1 / x {\displaystyle 1/x} (approximately 36.8 % {\displaystyle 36.8\%} when x = e {\displaystyle x=e}), so that essentially the same optimal division appears in optimal planning problems like the secretary problem.