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

In article <deif86$202$1@xxxxxxxxxxxxxxxxxxxxxx>,
israel@xxxxxxxxxxx (Robert Israel) wrote:

> 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.

So I think the bottom line is,

1) if you're a student in a calculus class, a point of inflection
is whatever your teacher says it is, and

2) if you're not a student in a calculus class, and for some reason
you feel compelled to make use of the concept of a point of inflection,
be sure you spell out exactly what you mean by it.

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.

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