Proof of derivatives through induction

• May 7th 2012, 04:26 AM
MarkJacob
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.

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 :)

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.