I am trying to find the name of the point at which the derivative of a function at that point is zero but it is not a local max or min within any interval. For instance, take the function $y=(x+1)(x-1)^3$. The roots of the first derivative are -0.5, one, and one but the function at $x=1$ is neither a local max or a local min. What is the name of this point? Is it just a Point of Inflection or is there another name?
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J. W. Tanner
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Logo 4000
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1'Point of Inflection' is pretty much the canonical name. – Steven Stadnicki Oct 28 '20 at 16:51
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Okay, thank you! – Logo 4000 Oct 28 '20 at 16:53
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A point of inflection is where the tangent line crosses the curve. – B. Goddard Oct 28 '20 at 16:56
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I have heard some people call it a "Plateau", but maybe that's just me – person Oct 28 '20 at 17:17
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Also called inflection point. – Brian Tung Oct 28 '20 at 17:17
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A colleague of mine from the late 1990s used the term terrace point for a point $x=a$ where the first derivative is zero and the first derivative doesn't change sign as you pass through $x=a,$ and I liked the idea so much that I subsequently used it in all my calculus classes. Of course, this was only used in simple introductory calculus examples where the zeros of the derivative are isolated from each other.
A few years later (in July 2002; see 6. TERRACE POINTS IN THE FIRST DERIVATIVE TEST here) I was told that the term "terrace point" is in Ostebee/Zorn's Calculus book, although I've never looked at a copy to be sure, and I don't know whether the term was in both the 1994 1st and 2002 2nd editions or only in the 2002 2nd edition.
Anyway, when I last wrote about this term (as far as I can recall), there wasn't nearly as much on the internet as there is now, and google-books searching wasn't available. A google search shows that the term "terrace point" is now fairly widely used. Also, a google-books search shows that it is definitely used in the 2002 2nd edition of Ostebee/Zorn's book, as well as in several other books. Interestingly, the term also appears on p. 40 (line −10) of William Richard Ransom's 1915 Early Calculus. As far as I can determine, the only use of "terrace point" in one of the math oriented Stack Exchanges is this answer from 24 October 2019.
One of the reasons I liked having a name for this this notion is that it allows you to label all four of the possibilities that can show up on a first derivative sign chart where the derivative is zero (and isolated from the other zeros):
$$ ++++0++++ \;\;\;\;\; \text {ter} $$
$$ ++++0---- \;\;\;\;\; \text {max} $$
$$ ----0++++ \;\;\;\;\; \text {min} $$
$$ ----0---- \;\;\;\;\; \text {ter} $$
Dave L. Renfro
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