Re: true or false: (x^5 - x) has inflexion point at origin, (x^4 - x) doesn't

In article <wcVU$LADIHDDFwZy@xxxxxxxxxxxxxxxxx>,
David Jones <djones@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:
>In article <KAZOe.664781$cg1.361423@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>N. Silver <mathelp@xxxxxxxxxxxxxxxx> writes
>>David Jones wrote:
>>> Gerry Myerson writes:
>>>> David Jones wrote:

>>>>> This is equivalent to saying that a point on a function is a point of
>>>>> inflexion only if the FIRST DERIVATIVE IS AT AN EXTREMUM.

>>>>A point of inflection is a point where the 2nd derivative changes sign.

>>> Hi Gerry, and thanks for your reply, but is my formulation a way of
>>> saying the same thing, or is it faulty in some way?

>>> The condition 'first derivative at an extremum' appears to be equivalent
>>> to the condition 'second derivative changes sign'.

>>> But maybe I've missed something?

>>Consider y = abs(x^2 - 4).
>>According to what Gerry writes there are
>>points of inflection at x = -2 and x = 2.
>>But the derivative is not defined at these
>>numbers.
>>.

>Excuse my ignorance, but would it normally be said that these sharp
>points are points of inflection?

There doesn't seem to be a consensus on this among authors of calculus
texts. For example:

Apostol: At a point (c, f(c)) where the derivative f'(c) exists ...
The point c is called a point of inflection of the graph if, in some
open interval about c, g(x) [ = f(x) - (f(c) + f'(c)(x-c)) ] is
positive on one side of c and negative on the other side of c.

Adams: We say that the point (x_0, f(x_0)) is an inflection point
of the curve y = f(x) ... if the following conditions are satisfied:
(a) the graph y = f(x) has a tangent line at x_0, and
(b) the concavity of f is opposite on opposite sides of x_0.

Stewart: A point P on a curve is called an inflection point if the
curve changes from concave upward to concave downward or from concave
downward to concave upward at P.

Stewart gives an example similar to y = abs(x^2 - 4) which he does
call an inflection. Adams and Apostol would not call these inflections.
On the other hand, Adams would allow a vertical tangent line (e.g.
y = x^(1/3) at x = 0) which Apostol apparently would not. And then
there are examples such as y = x^3 + x^4 sin(1/x), y(0) = 0, which
Apostol would say has an inflection at x = 0 but Adams and Stewart would
not.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

Relevant Pages

  • Re: true or false: (x^5 - x) has inflexion point at origin, (x^4 - x) doesnt
    ... > There doesn't seem to be a consensus on this among authors of calculus ... > The point c is called a point of inflection of the graph if, ... > curve changes from concave upward to concave downward or from concave ... Adams and Apostol would not call these inflections. ...
    (sci.math)
  • Re: Could a tetration function exist?
    ... once tried to determine where this inflection point must lie. ... the growth of yslows down until it crosses ... then it starts to speed up towards infinity. ... concave down without a point of inflection. ...
    (sci.math)
  • Re: point of inflection AND extreme point
    ... Depends on your exact definition of point of inflection. ... so it is concave down. ... --- Calvin ... Arturo Magidin ...
    (sci.math)