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 Post subject: True or False (proof or counterexample)Posted: Thu, 9 Feb 2012 22:41:22 UTC
 Senior Member

Joined: Sat, 10 Oct 2009 15:43:22 UTC
Posts: 98
Suppose that f' is continuous in a neighborhood of c and f'(c)>0. Then f is strictly increasing in some neighborhood of c.

I'm having trouble determining this from the information given. I feel as if I can give an obvious counterexample, but the "SOME neighborhood of c" part makes me wary. Any help appreciated.

_________________
let:
a=b
ab=b^2
ab-a^2=b^2-a^2
a(b-a)=(b+a)(b-a)
a=b+a
a=2a
1=2

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 Post subject: Re: True or False (proof or counterexample)Posted: Fri, 10 Feb 2012 00:51:30 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Fri, 1 Jul 2011 01:17:26 UTC
Posts: 322
mgfortes wrote:
Suppose that f' is continuous in a neighborhood of c and f'(c)>0. Then f is strictly increasing in some neighborhood of c.

I'm having trouble determining this from the information given. I feel as if I can give an obvious counterexample, but the "SOME neighborhood of c" part makes me wary. Any help appreciated.

Some neighborhood implies that the neighborhood be small enough that f'(x) > 0 in the neighborhood. There is such a neighborhood, since f'(c) > 0 and f'(x) is continuous.

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 Post subject: Re: True or False (proof or counterexample)Posted: Fri, 10 Feb 2012 00:54:07 UTC
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Joined: Wed, 21 May 2003 04:27:18 UTC
Posts: 992
mgfortes wrote:
Suppose that f' is continuous in a neighborhood of c and f'(c)>0. Then f is strictly increasing in some neighborhood of c.

I'm having trouble determining this from the information given. I feel as if I can give an obvious counterexample, but the "SOME neighborhood of c" part makes me wary. Any help appreciated.

Might this be relevant?

http://en.wikipedia.org/wiki/Darboux's_theorem_(analysis)

_________________
"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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 Post subject: Re: True or False (proof or counterexample)Posted: Fri, 10 Feb 2012 01:47:36 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
Justin wrote:
mgfortes wrote:
Suppose that f' is continuous in a neighborhood of c and f'(c)>0. Then f is strictly increasing in some neighborhood of c.

I'm having trouble determining this from the information given. I feel as if I can give an obvious counterexample, but the "SOME neighborhood of c" part makes me wary. Any help appreciated.

Might this be relevant?

http://en.wikipedia.org/wiki/Darboux's_theorem_(analysis)

Not particularly, as is noted in the article, if is continuous, then it just follows from the MVT, which is what you really want. Since is continuous, we know that the inverse image of an open set around is an open set surrounding , but then if we take , then this is clearly an open neighborhood of which contains only positive numbers since , so its inverse image is the desired neighborhood of .

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 Post subject: Re: True or False (proof or counterexample)Posted: Fri, 10 Feb 2012 01:51:29 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 21 May 2003 04:27:18 UTC
Posts: 992
Justin wrote:

Not particularly, as is noted in the article, if is continuous, then it just follows from the MVT, which is what you really want. Since is continuous, we know that the inverse image of an open set around is an open set surrounding , but then if we take , then this is clearly an open neighborhood of which contains only positive numbers since , so its inverse image is the desired neighborhood of .

Ok, that makes sense, the problem seemed to have an IVT ring to it, but I didn't know if that was strong enough...unlikely since was an explicit hypothesis. I will see just for kicks if I can work out a proof by contradiction.

_________________
"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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 Post subject: Re: True or False (proof or counterexample)Posted: Fri, 10 Feb 2012 01:57:47 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
Justin wrote:
Justin wrote:

Not particularly, as is noted in the article, if is continuous, then it just follows from the MVT, which is what you really want. Since is continuous, we know that the inverse image of an open set around is an open set surrounding , but then if we take , then this is clearly an open neighborhood of which contains only positive numbers since , so its inverse image is the desired neighborhood of .

Ok, that makes sense, the problem seemed to have an IVT ring to it, but I didn't know if that was strong enough...unlikely since was an explicit hypothesis. I will see just for kicks if I can work out a proof by contradiction.

Any direct proof can be made into a proof by contradiction, just assume you're wrong, and go through the usual proof, you arrive at the truth, which is the contradiction to the falsehood. True proofs by contradiction involve existence theorems which don't produce an explicit result. However, this should be possible for you, at least in theory, since the IVT and MVT are not constructive theorems.

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 Post subject: Re: True or False (proof or counterexample)Posted: Fri, 10 Feb 2012 02:40:35 UTC
 Senior Member

Joined: Sat, 10 Oct 2009 15:43:22 UTC
Posts: 98
Thank you both for your replies.

That was a fantastic insight and very helpful. Thanks!

_________________
let:
a=b
ab=b^2
ab-a^2=b^2-a^2
a(b-a)=(b+a)(b-a)
a=b+a
a=2a
1=2

Top

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