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Proving Inequalities

  1. Feb 22, 2009 #1
    I never have much luck when something says "prove something <= something else". I usually just fiddle around and occasionally get lucky and reduce it to a constant < an express I know can't be less than that constant. But most times I can't reduce it to something like that. Is there any kind of "principled" approach that someone can suggest/ point me to read about?

    As an example, say it was something like
    [tex]ln(1+a) \leq a^2 b[/tex]

    How would you approach something like this?

    Thanks,

    Dave
     
  2. jcsd
  3. Feb 22, 2009 #2
    You can, in general, use any number of arguments to show something like that.

    For the natural log one, just raise e to both sides. Then you get

    1 + a = exp(b*a^2)

    Then you can write

    1 = exp(b*a^2) - a.

    I don't know... then just pull out an exponential term and get

    1 = exp(b*a^2)(1 - a*exp(-b*a^2)) and argue that, as either A or b grow without bound, the right hand simplifies to exp(b*a^2) and the left remains constant. And for a, b both positive, this is clearly true.

    I mean, it's just algebra, some logic, and knowing what you're trying to show. Does that help?
     
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