> --- In Math4u@yahoogroups.com, "Brian Edward Jensen" wrote:
> >
> > Ray,
> > The function F(x)=x is a line having a slope of 1. This line goes on
> > in both directions forever. If there were ends to the line, the max
> > and min would be at the ends. But this line has no maximum or
> > minimum.
> Wrong again. F(x) = x reaches a maximun when x = infinity, and a
> minimum when x = minus infinity, by definition.
>
> By your logic, f(x) = x² would have no minima, and f(x) = -x² would
> have no maxima.
No, very much correct. Maxima and minima are reached at points. Points
have coordinates, which are real numbers. Infinity is not a real number,
therefore, m = (inf, inf) is not a point, and can therefore not be the
maximum of the fucntion F(x) = x.
Another way of looking at this is, if you'd define a function equal to
F(x) = x on some interval (let's say [-1, 1], because it's easy), and
remain undefined for all other values, then you'd give the line an end
(in this case (1, 1)) and the endpoint would be the maximum.
Also, you're reasoning about F(X) = x^2 is incorrect. Brian did *not*
say "there are no ends to this line, so it has no max", he said "there
are no ends to this line, so it can continue growing indefitely, and
therefore it has no max". F(x) = x^2 has no ends either, but it won't
grow to arbitrarily low values, so m = (0, 0) is a completely valid
minimum point.
Grtz,
Rob
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