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Math Help - Proof of derivatives through induction

  1. #1
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    Proof of derivatives through induction

    Hey guys,

    Have a problem that was set as exam revision and was told that there'd be a question similar on the exam, so I'm very keen on understanding how to do this problem well.

    The problem asks:

    Let f(x) = {e-1/x^2, if x =/= 0
    {0, if x = 0

    Prove that f(n)(0) = 0 for positive integer values of n.

    I've figured that induction would be a good way to go about solving this problem obviously, although I'm not too sure how to prove it for any case.
    Any help would be greatly appreciated

    Thanks in advance,
    Mark
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  2. #2
    MHF Contributor
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    Re: Proof of derivatives through induction

    You need the fact that \lim_{t\to\infty}e^{-t^2}t^n=0 for all n.

    Prove by induction on n that there exists a polynomial p_n such that f^{(n)}(x)=e^{-1/x^2}p_n(1/x) for x\ne 0 and f^{(n)}(0) = 0. The latter part for n + 1 is proved by the definition of the derivative using the induction hypothesis.
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