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Are polynomials infinitely many times differentiable?

If so, does it only mean that at some point we reach 0 and then we keep on getting 0?

Thank you!

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    $\begingroup$ That is exactly correct. Good thinking! $\endgroup$ Jun 29, 2015 at 22:33
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    $\begingroup$ Yes, polynomials are infinitely many times differentiable, and yes, after some finite number of derivatives (specifically $\deg f + 1$) we get $0$, and then we continue to get $0$ thereafter. $\endgroup$
    – mweiss
    Jun 29, 2015 at 22:33
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    $\begingroup$ ... but that's not what it means, it's just what happens in this case when you keep differentiating. $\endgroup$ Jun 29, 2015 at 22:34
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    $\begingroup$ For example, $\sin x$ is infinitely differentiable, but you never reach and stay at zero. $\endgroup$ Jun 29, 2015 at 22:36

2 Answers 2

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Yes.

$$\frac{d}{dx} 0 = 0,$$ so once you've taken $\deg(p) + 1$ derivatives, where $p$ is the polynomial you're considering, you will just keep getting zero.

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    $\begingroup$ If in this logic, then constant function is also infinitely differentiable? $\endgroup$
    – Userkkr
    Apr 5, 2017 at 1:40
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    $\begingroup$ Yes --- the constant functions are just a special case of this, being the degree zero polynomials. $\endgroup$
    – qaphla
    Apr 10, 2017 at 21:03
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The other answers probably answered your question, but in case you wanted to learn more, the key word to look up is smoothness of the function (see here for instance: https://en.wikipedia.org/wiki/Smoothness).

So another, more fancy way of asking your question is whether or not polynomials are smooth functions.

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  • $\begingroup$ Only one of the two other answers is above. You may want to change your wording to make it more independent of surroundings. $\endgroup$
    – Ruslan
    Jun 30, 2015 at 10:28
  • $\begingroup$ Changed the wording. Thanks! $\endgroup$ Jun 30, 2015 at 13:32

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