Re: infinitely differentiable functions

On Thu, 08 Nov 2007 02:35:29 -0500, quasi <quasi@xxxxxxxx> wrote:

Let f be a C^infinity function from R to R such that f and all its
derivatives are everywhere nonzero.

Two questions, probably easy ...

(1) If f is everywhere positive, must all derivatives of f also be
everywhere positive?

(2) If the answer to question (1) is no, can we have an arbitrary
sequence of signs for f, f', f'', ... ?

Ok, question (1) is easy.

The answer is "no" -- for example, f(x) = e^(-x).

quasi
.

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