Last night an old idea came back to me, an idea about a function where all the derivatives start from zero and then grow smoothly. I thought it would be impossible, but then I found some interesting stuff on Wikipedia. So, I learned to use SymPy and wasted a lot of time with it. Here's a report of my (non-)findings.
(UPDATE: I did some numerical differentiation, which showed that h(x) does have negative derivatives. See details in this comment. A disappointment, although perhaps not a surprising one. It doesn't however, necessarily mean the goal is impossible.)
So, if anyone knows whether such a function exists and what it looks like, please tell me.
I think you need to have a discontinuity in a derivative at some level to have a function like this where the lower derivatives grow smoothly. If you have zero at all levels and no discontinuities... nothing should ever change, right?
That's what I thought too, but it turns out that is not true! Here's a couple of interesting links: Non-analytic smooth function, Bump function, Flat function. EDIT: Fixed the links and added another one.
Thanks for the links. This is outside the area of math I usually deal with, but I agree it's interesting. I think I understand what you're asking for now, but I've hit my mental limit for today trying to work it out. Good luck in your search!
f(x) = e^(-1/x^2) for x != 0, f(0) = 0. It's relatively easy to show this is infinitely differentiable at x=0 and every derivative is 0.
The intuition that an infinitely differentiable function can be described globally by its derivatives locally is actually true for complex differentiable functions, and this property is sometimes referred to as "rigidity" of complex-differentiable (or analytic/holomorphic) functions. It doesn't hold for functions that are only differentiable along the real axis.