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In article
<1162284539.979664.169410@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"smn" <smnewberger@xxxxxxxxxxx> wrote:
> The World Wide Wade wrote:
> > In article
> > <25039834.1162267012389.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
> > craig <ctcowan@xxxxxxxxxxx> wrote:
> >
> > > Does there exist a smooth, non-zero, compactly supported function u on R
> > > such
> > > that
> > >
> > > v(x) (v defined below) can be extended to a smooth function on R. ??
> > >
> > >
> > > Let U denote the open set where u does not equal zero and define
> > >
> > > v(x):= (u''(x))/ u(x) on U.
> >
> > Hint: Suppose u(x) = 0 for x <= 0 and u(x) is nonzero for small x
> > > 0. For such x, use the mean value theorem twice to see
> > |u''(x)/u(x)| >= |u''(x)/[u''(c_x)*x^2]|. That indicates v is
> > blowing up like 1/x^2 near 0.
> u''(x)/u''(c_x) may tend to 0 so I don't see your hint as conclusive.smn
A hint is not supposed to be conclusive, although it should be
helpful. And I think this one is.
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