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?

1 Answer
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(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.