Could you please give me some hint how to solve this problem:
Suppose $f(x)$ is infinitely many times differentiable function on R, $f(0)=f'(0)=f''(0)=0$.
Prove : for all $A>0$ exists some $c>0$ such as $|f(x)|\le c{|x|}^3$ for all $x\in[-A,A]$.
I could not figure out how to start.
Thanks.
How about the Taylor formula:
$$f(x) = f(0) + f'(0)x + 1/2 f''(0)x^{2} + 1/3! f'''(\xi)x^{3}$$
Besides, f(x) is infinitely many times differentiable function on R.
s.t. $f'''(x)$ is continuous.