Question

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?

Answer 1

.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(c,f(c))` on the graph of a function `f` is a point of inflection if there exists an open interval `(a,b)` containing `c` such that the following relations hold:

(i)`color(white)(')` `" "` `f''(x) > 0` if `a < x < c` and `f''(x) < 0` if `c < x < b`; or
(ii)`" "` `f''(x) < 0` if `a < x < c` and `f''(x) > 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(x) = {(tanx",",x < 0),(tanx+2",",x >= 0) :}`

has inflection point `(0,2)`. and

`g(x) = {(tanx",",x <= 0),(tanx+2",",x > 0) :}`

has inflection point `(0,0)`.

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