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Functions with increasing derivatives

  1. Nov 18, 2013 #1
    Consider a function f(x), such that for all points x0 in the domain, the nth derivative of f evaluated that x0 is less than the n+1th derivative of f evaluated at x0.

    A quick example is f(x) = e^(ax) where a > 1, what others are there (not including just changing e to something else)? Is there a name for these?
  2. jcsd
  3. Nov 18, 2013 #2


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    It's easy to invent infinite power series that have this property. For example ##f(x) = 0 + x + 2x^2 + 3x^3 + \dots##.

    Proving (a) it has the required property and (b) it is convergent (for some real values of ##x##) are left as exercises for the OP.

    Actually, you don't need the infinite series. ##f(x) = -1/x##, when ##-1 < x < 0##.
    Last edited: Nov 18, 2013
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