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.

It's true whenever $f$ is twice differentiable at $0$.
In that case, $$\displaystyle f(x)=f''(0)\frac{x^2}{2}+o(x^2)=f''(0)\frac{x^2}{2}+x^2\epsilon(x^2)$$ where $\epsilon$ is a function such that $\lim_{x\to 0}\epsilon(x)=0$
Consequently, there exists some $\delta >0$ such that $|x|\leq \delta\implies |\epsilon(x^2)|\leq \left|\frac{f''(0)}{4}\right| $
Reverse triangular inequality implies $|f(x)|\geq \left|\frac{f''(0)}{4}\right|x^2$ whenever $ |x|\leq \delta$.
