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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.

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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.

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  • $\begingroup$ There exist infinitely differentiable functions which do not equal their Taylor series - see here on Wikipedia. $\endgroup$ – complexist May 23 '14 at 14:07
  • $\begingroup$ @complexist I definitely think this function is not equal to their Taylor series, but Taylor formula still work. $\endgroup$ – Eric May 26 '14 at 9:56

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