# Is there a way to find the critical points and inflection points of a finction w/o derivitives?

**URL:** <https://boards.straightdope.com/t/is-there-a-way-to-find-the-critical-points-and-inflection-points-of-a-finction-w-o-derivitives/540743>\
**Category:** Factual Questions\
**Created:** [May 25, 2010, 8:12pm UTC](https://boards.straightdope.com/t/is-there-a-way-to-find-the-critical-points-and-inflection-points-of-a-finction-w-o-derivitives/540743 "2010-05-25T20:12:27Z")\
**Posts on this page:** 3\
**Page:** 2

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**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [May 26, 2010, 2:51am UTC](https://boards.straightdope.com/t/is-there-a-way-to-find-the-critical-points-and-inflection-points-of-a-finction-w-o-derivitives/540743/21 "2010-05-26T02:51:21Z")

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I still don’t understand. How is taking the integral useful? You’ve moving from a function whose second derivative’s changes of sign are the points of interest to a function whose third derivative’s changes of sign are the points of interest. It’s all the bother of taking the derivative, except in the other direction, further away from what you’re looking for!

For example, consider the function 4x[sup]3[/sup] + 9x[sup]2[/sup] + 1. Suppose I want to know where its inflection point is and I start by taking the integral, x[sup]4[/sup] + 3x[sup]3[/sup] + x + C. Now what?

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [May 26, 2010, 4:05am UTC](https://boards.straightdope.com/t/is-there-a-way-to-find-the-critical-points-and-inflection-points-of-a-finction-w-o-derivitives/540743/22 "2010-05-26T04:05:17Z")

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> [@Hypnagogic\_Jerk](#):
>
> I believe I’ve seen an extension of critical points to functions from **R** [sup]n[/sup] to **R** [sup]m[/sup], but it’s been a long time so I don’t remember.

Check the Wiki article I linked to.

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**Author:** ![GameHat](https://avatars.discourse-cdn.com/v4/letter/g/ed655f/32.png) [@GameHat](https://boards.straightdope.com/u/GameHat)\
**Post date:** [May 27, 2010, 12:37am UTC](https://boards.straightdope.com/t/is-there-a-way-to-find-the-critical-points-and-inflection-points-of-a-finction-w-o-derivitives/540743/23 "2010-05-27T00:37:46Z")

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> [@Ace309](#):
>
> The OP did, when he asked if graphing was the only alternative.

Ah, my mistake. It was late.

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