# Easy calculus question ... **URL:** \ **Category:** Factual Questions\ **Created:** [March 31, 2003, 6:33pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501 "2003-03-31T18:33:55Z")\ **Posts on this page:** 15\ **Page:** 1
**Author:** ![Greg\_Charles](https://avatars.discourse-cdn.com/v4/letter/g/839c29/32.png) [@Greg\_Charles](https://boards.straightdope.com/u/Greg_Charles)\ **Post date:** [March 31, 2003, 6:33pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/1 "2003-03-31T18:33:55Z")
It’s been too long since I studied this stuff … If a function is increasing, then comes to a point where the slope (first derivative) is 0, then decreases, that point is a local maximum. If the function decreases, then reaches a 0 slope and begin to increase, that point is a local mininum. The other possibility is the function is increasing, reaches a point where the slope is 0, and then keeps increasing. (Or vice versa.) In this case, both the first and second derivatives are 0 at that point. What is that point called? ---
**Author:** ![Firx](https://avatars.discourse-cdn.com/v4/letter/f/c2a13f/32.png) [@Firx](https://boards.straightdope.com/u/Firx)\ **Post date:** [March 31, 2003, 6:37pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/2 "2003-03-31T18:37:22Z")
A point of inflection. ---
**Author:** ![Joe\_Cool](https://avatars.discourse-cdn.com/v4/letter/j/ecb155/32.png) [@Joe\_Cool](https://boards.straightdope.com/u/Joe_Cool)\ **Post date:** [March 31, 2003, 6:37pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/3 "2003-03-31T18:37:23Z")
Inflection points. ---
**Author:** ![tool462](https://avatars.discourse-cdn.com/v4/letter/t/e5b9ba/32.png) [@tool462](https://boards.straightdope.com/u/tool462)\ **Post date:** [March 31, 2003, 6:49pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/4 "2003-03-31T18:49:26Z")
An inflection point is any point where the second derivative is zero. (i.e., sin(x) at x=0) When both the first and second derivative are zero, it is called a terrace point. (i.e., x^3 at x=0) ---
**Author:** ![tool462](https://avatars.discourse-cdn.com/v4/letter/t/e5b9ba/32.png) [@tool462](https://boards.straightdope.com/u/tool462)\ **Post date:** [March 31, 2003, 6:51pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/5 "2003-03-31T18:51:53Z")
Follow up: apparently it is also called a stationary point. [http://mathworld.wolfram.com/FirstDerivativeTest.html](http://mathworld.wolfram.com/FirstDerivativeTest.html) ---
**Author:** ![Greg\_Charles](https://avatars.discourse-cdn.com/v4/letter/g/839c29/32.png) [@Greg\_Charles](https://boards.straightdope.com/u/Greg_Charles)\ **Post date:** [March 31, 2003, 7:22pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/6 "2003-03-31T19:22:38Z")
Hmm … this question may be harder than I thought. According to tool462’s link: “A stationary point may be a minimum, maximum, or inflection point.” That means when the first derivative is 0, it’s called a stationary point. When the second derivative is 0, it’s an inflection point. (Thanks to firx and Joe\_Cool, but I did remember that much.) What is it called when both become 0? A terrace point seems plausible, but I’ve never heard that term, nor can I find any confirmation of it. My searches keep turning up info on mathematics programs in communities called “Terrace Point”. ---
**Author:** ![tool462](https://avatars.discourse-cdn.com/v4/letter/t/e5b9ba/32.png) [@tool462](https://boards.straightdope.com/u/tool462)\ **Post date:** [March 31, 2003, 7:28pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/7 "2003-03-31T19:28:08Z")
Found another link on MathWorld that may answer your question: [http://mathworld.wolfram.com/SaddlePoint.html](http://mathworld.wolfram.com/SaddlePoint.html) I was under the (mistaken) impression that saddle points were in two dimensional functions (look at a graph of z = x^2 - y^2 to see why). It appears saddle point is the more accepted term for this than terrace point. Hope this helps. ---
**Author:** ![Yumanite](https://avatars.discourse-cdn.com/v4/letter/y/b487fb/32.png) [@Yumanite](https://boards.straightdope.com/u/Yumanite)\ **Post date:** [March 31, 2003, 7:45pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/8 "2003-03-31T19:45:24Z")
I remember calling it a saddle point in my calc class. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [March 31, 2003, 7:54pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/9 "2003-03-31T19:54:16Z")
Not all points where the first and second derivatives are zero are terrace points, either. Consider y=x[sup]4[/sup], at the origin. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [March 31, 2003, 8:42pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/10 "2003-03-31T20:42:53Z")
Nitpick for **tool462** : At least according to the calculus books I’ve seen, an inflection point is where the concavity changes. This is usually but not always the same as where the second derivative is zero. For example, y = x^4 has second derivative = 0 at (0,0) but is concave upward everywhere else; while y = x^(1/3) changes concavity at (0, 0) but has an undefined second derivative there. ---
**Author:** ![tool462](https://avatars.discourse-cdn.com/v4/letter/t/e5b9ba/32.png) [@tool462](https://boards.straightdope.com/u/tool462)\ **Post date:** [March 31, 2003, 9:08pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/11 "2003-03-31T21:08:42Z")
Good points, Chronos & Thudlow. I didn’t think about those cases. According the MathWorld link above, there is a third condition for the saddle point that the third derivative is non-zero, which takes care of y = x^4. I’m not sure what to make of x^1/3. I’m guessing, though, that saddle point is what Greg was looking for. ---
**Author:** ![Trigonal\_Planar](https://avatars.discourse-cdn.com/v4/letter/t/0ea827/32.png) [@Trigonal\_Planar](https://boards.straightdope.com/u/Trigonal_Planar)\ **Post date:** [March 31, 2003, 9:32pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/12 "2003-03-31T21:32:06Z")
I also thought they were inflection points - changes in concavity. ---
**Author:** ![asterion](https://avatars.discourse-cdn.com/v4/letter/a/4da419/32.png) [@asterion](https://boards.straightdope.com/u/asterion)\ **Post date:** [March 31, 2003, 9:36pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/13 "2003-03-31T21:36:17Z")
> [@](#): > > \*Originally posted by Yumanite \* > \*\*I remember calling it a saddle point in my calc class. \*\* I believe that term is only used in multivariable calc, where you’re looking at a surface. I have never heard saddle point used in two dimensions. ---
**Author:** ![Greg\_Charles](https://avatars.discourse-cdn.com/v4/letter/g/839c29/32.png) [@Greg\_Charles](https://boards.straightdope.com/u/Greg_Charles)\ **Post date:** [March 31, 2003, 10:12pm UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/14 "2003-03-31T22:12:53Z")
I’m with asterion here. Saddle point is what jumped into my head at first, but my recollection of saddle point is a point that is a local minimum of one partial derivative and local maximum of the other for some f(x,y). The graph of that function would have to look something like a saddle to have a saddle point. tool’s link gainsays my memory though, saying that what I asked for in the OP is, in fact, also called a saddle point. I didn’t realize that when the third derivative is zero, that the “inflection points” are not really inflection points. It makes sense though. I still haven’t found a working definition of terrace point, so I can’t say anything about Chronos’s claim that (0,0) is not a terrace point for f(x) = x^4. It’s apparently not an inflection point though. However, that leaves some ambiguity at tool’s math site, since it claims a stationary point is anywhere the first derivative vanishes. (I assume that means goes to zero.) It says a stationary point can be a local maximum, local minimum, or inflection point. Yet, here we have an example of the first derivative being zero where none of those definitions fit. ---
**Author:** ![g8rguy](https://avatars.discourse-cdn.com/v4/letter/g/7cd45c/32.png) [@g8rguy](https://boards.straightdope.com/u/g8rguy)\ **Post date:** [April 1, 2003, 12:50am UTC](https://boards.straightdope.com/t/easy-calculus-question/165501/15 "2003-04-01T00:50:07Z")
from the [same](http://mathworld.wolfram.com/SaddlePoint.html) site: “A point of a function or surface which is a stationary point but not an extremum. An example of a one-dimensional function with a saddle point is f(x)=x[sup]3[/sup].”