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

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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?

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?

##### 1 Answer

.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 *point of inflection* if there exists an open interval

(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 **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

has inflection point

has inflection point

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