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Topic: [ap-calculus] Re: local minimum question
Replies: 1   Last Post: Jan 31, 2005 9:58 AM

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Richard J Maher

Posts: 80
Registered: 12/6/04
[ap-calculus] Re: local minimum question
Posted: Jan 31, 2005 9:58 AM
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Hello everyone,

On Thu, 27 Jan 2005 wrote:

... I, and others, to quote Mark Howell's phrase, like
to think of local minimum (and maximum) as the smallest
(largest) value in the range "around here." Thus the
endpoint of a closed interval could be, and in fact must
be, a local maximum or minimum. ...

We have to be a little careful here. "Nice" continuous functions
have either a local minimum or a local maximum at the endpoints of
a closed interval. However, the function defined on [0,1] by

f(x) = xsin(1/x) for x in (0,1] while f(0)=0

is continuous on [0,1] but does not have either a local minimum or
a local maximum at x=0. It is not too hard to see this, since for
x>0, sin(1/x) takes on the values -1 and 1 on any interval of the form
(0,d) for any d<1. As a result, xsin(1/x) takes on both positive and
negative values no matter how close we get to zero.

Hope this helps,

Dick Maher

Richard J. Maher
Mathematics and Statistics
Loyola University Chicago
6525 N. Sheridan Rd.
Chicago, Illinois 60626

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