Is there any function f : R + → R + satisfying the following conditions?:the sequence f ( n ) is convergent; log ⁡ f is eventually monotone (i.e., it is monotone from a number on); log ⁡ f is not eventually convex or concave.If yes, is there a (big) class of such functions (with various properties)?

Question

Is there any function   f  :            R        +    →            R        +   satisfying the following conditions?:the sequence   f  (  n  ) is convergent;  log  ⁡  f is eventually monotone (i.e., it is monotone from a number on);  log  ⁡  f is not eventually convex or concave.If yes, is there a (big) class of such functions (with various properties)?

cesantedz · Accepted Answer

There are lots of such functions. Let g be a positive increasing function on             R              +       such that   g  (  n  )  =  1  −  1      /    n for each   n and such that   g does not have a left derivative at some point in   (  k  ,  k  +  1  ) for each   k. Let   f  =  e  g. Then   l  o  g  f is not concave or convex eventually because convex and concave functions have left derivatives at every point. Of course, one can use right derivatives instead of left derivatives.

Is there any function f:R^+->R^+ satisfying the following conditions? the sequence f(n) is convergent; log f is eventually monotone (i.e., it is monotone from a number on); log f is not eventually convex or concave. If yes, is there a (big) class of such functions (with various properties)?

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New Questions in Calculus 1