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

Re: Proof of derivatives through induction

You need the fact that for all n.

Prove by induction on n that there exists a polynomial such that for and . The latter part for n + 1 is proved by the definition of the derivative using the induction hypothesis.