Finding a differentiable function that cannot be bounded by quadratic function Is there any function from $\mathbb R$ to $\mathbb R$ such that it's value at $0$ is $0$ and its derivative at $0$ is also $0$ and can never be bounded by any form of quadratic function in any neighborhood of the origin? 
I have been thinking about this for a long time but got no clue.

What about $f(x) = |x|^{3/2}$?
It satisfies your condition, but $\lim_{x\to 0} \dfrac{f(x)}{x^2} = \infty$, so $f(x)$ is larger than $cx^2$ near $x=0$ for every $c>0$. 
(You may have another kind of "bound" in mind though.)
