I was unfamiliar with the way a function and its limits are written. I thought the function was all sine wave not realizing that we are talking about a sine wave with the bottom of one or two waves cut off.
The thought hit me that the maxima would be points and perhaps we can call the minima the horizontal line segments. Perhaps this is what Steven Tavener said.
So I looked up the definition of maximum and minimum.
http://www.math.com/tables/derivatives/extrema.htm
It looks to me from the definition that the horizontal segment can be called a minimum. If so, then if there are 2 maxima on a continuous curve, then there must be a minimum between them.If you define a minimum as single point, then there is no minimum. I want to be rich and many other things and absolutely don't care about this.
Regards,
Brian Jensen
__._,_.___----- Original Message -----From: sanjiva dayalSent: Saturday, October 06, 2007 11:58 PMSubject: Re: RE: [Math4u] Re: Maxima & MinimaHi,
1.
Brian, there is no minima at 3pi/2 and 7pi/2.
2.
A function has minima at 'a' if f(x)>f(a) (not
f(x)>=f(a)) in a neighbourhood of 'a', x not equal to
'a'.
Sanjiva
--- Rob van Wijk <robvanwijk@gmx.net > wrote:
>
> http://mathworld.wolfram.com/ Minimum.html
> The smallest value of a set, function, etc.
> -> Note that no requirement is made this smallest
> value is unique
>
> http://en.wikipedia.org/wiki/ Maxima_and_ minima
> [A] function has a local minimum point at x'
> if f(x') <= f(x) when |x รขˆ' x'| < epsilon
> -> Note the use of smaller-or-equal (instead of
> strictly smaller)
>
> So, the function constructed by Sanjiva has a
> minimum
> {(4x-1)pi/2, for x in Integer} Union [3pi/2, 7pi/2]
> (the union of an infinite set of discrete points and
> a closed interval).
>
> While I haven't seen a formal prove yet, it
> certainly seems like
> the original proposition (the one Sanjiva's function
> was intended
> to disprove) holds.
>
> Grtz,
> Rob
>
>
> -------- Original-Nachricht --------
> > Datum: Wed, 3 Oct 2007 09:43:16 +0100
> > Von: "Stephen Tavener" <stephen.tavener@bbc.co.uk >
> > An: Math4u@yahoogroups.com
> > Betreff: RE: [Math4u] Re: Maxima & Minima
>
> > I think the point is more to do with definitions
> than anything else.
> >
> > If I understand correctly, according to Sanjeev:
> > At a local maximum, you can choose a small area
> around the point, where
> > that point is higher than every other point in the
> set.
> > At a local minimum, you can choose a small area
> around the point, where
> > that point is lower than every other point in the
> set.
> >
> > So, Sanjeev constructed a case where there is a
> plateau - no point is
> > lower than its' neighbour. So, while it is true
> that between two local
> > maxima, there are points with a minimum value,
> there does not need to be
> > a local minimum.
> >
> > However, as to whether this is a valid definition
> or not, I leave to the
> > reader. I reckon Wikipedia's and Mathworld's
> definitions would allow
> > all points in a plateau to be minima,
> contradicting Sanjeev, but in
> > maths, you start with a set of lies, and build
> elaborate structures, so
> > it just comes down to what lies we tell ourselves!
>
> >
> >
> > _____________________ _________ __
> >
> > From: Math4u@yahoogroups.com
> [mailto:Math4u@yahoogroups.com ] On Behalf
> > Of Brian E. Jensen
> > Sent: 02 October 2007 20:06
> > To: Math4u@yahoogroups.com
> > Subject: Re: [Math4u] Re: Maxima & Minima
> >
> >
> > max at (pi/2,1)
> > min at (3pi/2,-1)
> > max at (5pi/2,1)
> > min at (7pi/2, -1)
> > max at 9pi/2,1)
> > It looks to me like the statement is proved true
> by Sanjiva's example.
> > regards, Brian
> >
> >
> > ----- Original Message -----
> > From: sanjiva dayal
> <mailto:sanjivadayal@yahoo.com >
> > To: Math4u@yahoogroups.com
> <mailto:Math4u@yahoogroups.com >
> > Sent: Tuesday, October 02, 2007 3:42 AM
> > Subject: RE: [Math4u] Re: Maxima & Minima
> >
> >
> > Hi,
> > 1.
> > The statement of my question is proved false by
> the
> > following example:-
> > f(x)=sinx, x<=3*pi/2
> > =-1, 3*pi/2<x<7*pi/2
> > =sinx, x>=7*pi/2.
> > This function f(x) is defined for all real x, is
> > continuous & differentiable for all real x and
> has
> > local maximas at x=pi/2 and x=9*pi/2, but there
> is no
> > local minima betweem pi/2 and 9*pi/2. Therefore,
> the
> > initial statement is FALSE.
> > 2.
> > Moral of the story:- "Never use common sense in
> > Mathematics, always think from the definitions".
> > Sanjiva
> >
> > --- Stephen Tavener <stephen.tavener@bbc.co.uk
> > <mailto:stephen.tavener%40bbc. co.uk> > wrote:
> >
> > > Hmmm... what about a function which has a
> plateau
> > > between two local
> > > maximae?
> > >
> > > At this late remove, I can't remember all my
> > > definitions, but I think we
> > > can get away with a function like:
> > >
> > > f(x) = {
> > > g(x) for x <= k1
> > > h(x) for k1 < x <= k2
> > > i(x) for k2 < x
> > > }
> > >
> > > define h(x) to be 0 at all values, and choose
> > > suitable cubic equations
> > > for g(x) and i(x) such that g(k1) = 0, i(k2) =
> 0,
> > > g'(k1) = 0, i'(k2) =
> > > 0, and g(x) and i(x) have suitable maximae.
> > >
> > > If memory serves, this should meet the
> definitions
> > > of continuous and
> > > differentiable at all points, but somebody who
> has
> > > done a maths degree
> > > more recently than 20 years ago (ouch) can tell
> me
> > > if I'm wrong!
> > >
> > >
> > > --
> > > CH3 Stephen Tavener
> > > | DigiText Programmer, BBC News,
> > > New Media.
> > > N
> > > / \ Room BC3 xtn (020 800)84739
> > > N----C C==O Broadcast Centre, 201 Wood Lane,
> > > London. W12 7TP
> > > || || |
> > > || || | mailto:Stephen.Tavener@bbc.co.uk
> > <mailto:Stephen.Tavener%40bbc. co.uk>
> > > (Work)
> > > CH C N--CH3 mailto:mrraow@gmail.com
> > <mailto:mrraow%40gmail.com> (Home)
> > > \ / \ /
> > > N C http://www.scat.demon.co. uk/
> > <http://www.scat.demon.co. >uk/
> > > (Games)
> > > | ||
> > > CH3 O Baby pictures:
> > >
> >
>
http://rainbot.pwp.blueyonder. co.uk/Katiefrog/ group.jpg
> >
>
<http://rainbot.pwp.blueyonder. >co.uk/Katiefrog/ group.jpg
>
> > >
> > >
> > >
> > >
> > > _____________________ _________ __
> > >
> > > From: Math4u@yahoogroups.com
> <mailto:Math4u%40yahoogroups. com>
> >
> > > [mailto:Math4u@yahoogroups.com
> > <mailto:Math4u%40yahoogroups. com> ] On Behalf
> > > Of Brian E. Jensen
> > > Sent: 01 October 2007 17:38
> > > To: Math4u@yahoogroups.com
> <mailto:Math4u%40yahoogroups. com>
> > > Subject: Re: [Math4u] Re: Maxima & Minima
> > >
> > >
> > >
> > > I don't have time to figure out what Jim
> FitzSimons,
> > > who is very
>
=== message truncated ===
Sanjiva Dayal, B.Tech.(I.I.T. Kanpur)
Address:A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA.
Phones:+91-512-2581532,2581426.
Mobile:9415134052
Business email:sanjivadayal@yahoo.com
Personal email:sanjivadayal@hotmail.com
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