# What is the definition of inflection point? Or is it just not standarized like 0 in NN?

## The definition mentioned on socratic here and here defines it (in other words) as the point, where function is continuous and concavity changes sign. It is supported by a book. On the other side wiki says: "Inflection points are the points of the curve where the curvature changes its sign while a tangent exists." It is supported by another book. So, what's the deal with that tangent line?

Dec 31, 2017

.I think that it is not standardized.

#### Explanation:

As a student at a University in the US in 1975 we use Calculus by Earl Swokowski (first edition).

His definition is:
A point $P \left(c , f \left(c\right)\right)$ on the graph of a function $f$ is a point of inflection if there exists an open interval $\left(a , b\right)$ containing $c$ such that the following relations hold:

(i)$\textcolor{w h i t e}{'}$ $\text{ }$ $f ' ' \left(x\right) > 0$ if $a < x < c$ and $f ' ' \left(x\right) < 0$ if $c < x < b$; or
(ii)$\text{ }$ $f ' ' \left(x\right) < 0$ if $a < x < c$ and $f ' ' \left(x\right) > 0$ if $c < x < b$.
(pg 146)

In a textbook I use to teach, I think that Stewart is wise to include the condition that $f$ must be continuous at $c$ to avoid piecewise oddities. (See Note below.)

This is essentially the first alternative you mention. It has been similar in every textbook I have been assigned to use for teaching since then. (I have taught in several places in the US.)

Since joining Socratic I have been exposed to mathematicians who use a different definition for inflection point. So It appears that the usage is not universally defined.

At Socratic when answering questions about inflection points I usually state the definition as it appears in the question.

Note

Under Swokowski's definition, the function

$f \left(x\right) = \left\{\begin{matrix}\tan x \text{ & " & x < 0 \\ tanx+2" & } & x \ge 0\end{matrix}\right.$

has inflection point $\left(0 , 2\right)$. and

$g \left(x\right) = \left\{\begin{matrix}\tan x \text{ & " & x <= 0 \\ tanx+2" & } & x > 0\end{matrix}\right.$

has inflection point $\left(0 , 0\right)$.

Using Stewart's definition, neither of these functions has an inflection point.