I realise this is not calculus, yet still I'd appreciate if someone could help me

I have an entire function, f

and I need to show that the following is NOT true for ALL n=1,2,....

|f^{(n)}(0)| >= n^{n}*n!

(ie, the modulus of the nth derivative of f at zero is greater or equal to n to the power of n, times n factorial)

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so, my first attempt was by counter example, I picked a simple polynomial and showed the above expression is not true for all n.

however, I'm not sure if the question just wants a counter example, or a generalised proof for all such entire functions. If so, I'm slightly stumped... the only thought I have in this case is to look at Taylor series, but what will that yield?