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 − 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
> > intelligent said. But the initial statement is true:
> >
> >
> > If a function is differentiable over the whole
> > length, then you can't
> > have two maximums without having a minimum between
> > and you can't have
> > two minimums without having a maximum between.
> >
> > More generally, the function only needs to be
> > continuous for this to be
> > true. Not all continuous functions are
> > differentiable, but all
> > differentiable functions are continuous, for
> > example:
> >
> > y = absolute value of x.
> >
> > Regards, Brian
> >
> >
> >
> > ----- Original Message -----
> > From: w7anf <mailto:cherry@getnet.net
> <mailto:cherry%40getnet.net> >
> > To: Math4u@yahoogroups.com <mailto:Math4u%40yahoogroups.com>
> > <mailto:Math4u@yahoogroups.com
> <mailto:Math4u%40yahoogroups.com> >
> > Sent: Sunday, September 30, 2007 8:49 PM
> > Subject: [Math4u] Re: Maxima & Minima
> >
> >
> > A real function is said to be differentiable at a
> > point if its
> > derivative
> > exists at that point. The notion of
> > differentiability can also
> > be
> > extended
> > to complex functions (leading to the Cauchy-Riemann
> > equations
> > and
> > the theory
> > of holomorphic functions), although a few
> > additional subtleties
> > arise in
> > complex differentiability that are not present in
> > the real case.
> >
> >
> > Amazingly, there exist continuous functions which
> > are nowhere
> > differentiable.
> > Two examples are the Blancmange function and
> > Weierstrass
> > function.
> > Hermite
> > (1893) is said to have opined, "I turn away with
> > fright and
> > horror
> > from this
> > lamentable evil of functions which do not have
> > derivatives"
> > (Kline 1990, p. 973).
> >
> > The derivative of a function f(x) with respect to
> > the variable x
> > is
> > defined as
> >
> > f'(x)=lim((f(x+h)-f(x))/h,h,0) (6)
> >
> > but may also be calculated more symmetrically as
> >
> > f'(x)=lim((f(x+h)-f(x-h))/(2*h),h,0) (7)
> >
> > provided the derivative is known to exist.
> >
> > It should be noted that the above definitions refer
> > to "real"
> > derivatives,
> > i.e., derivatives which are restricted to
> > directions along the
> > real
> > axis.
> > However, this restriction is artificial, and
> > derivatives are
> > most
> > naturally
> > defined in the complex plane, where they are
> > sometimes
> > explicitly
> > referred
> > to as complex derivatives. In order for complex
> > derivatives to
> > exist, the
> > same result must be obtained for derivatives taken
> > in any
> > direction
> > in the
> > complex plane. Somewhat surprisingly, almost all of
> > the
> > important
> > functions
> > in mathematics satisfy this property, which is
> > equivalent to
> > saying
> > that
> > they satisfy the Cauchy-Riemann equations.
> >
> > These considerations can lead to confusion for
> > students because
> > elementary
> > calculus texts commonly consider only "real"
> > derivatives, never
> > alluding to
> > the existence of complex derivatives, variables, or
> > functions.
> > For
> > example,
> > textbook examples to the contrary, the "derivative"
> >
> > (read: complex derivative) d abs(z)/dz of the
> > absolute value
> > function abs(z)
> > does not exist because at every point in the
> > complex plane, the
> > value of the
> > derivative depends on the direction in which the
> > derivative is
> > taken
> > (so the
> > Cauchy-Riemann equations cannot and do not hold).
> > However, the
> > real
> > derivative (i.e., restricting the derivative to
> > directions along
> > the
> > real
> > axis) can be defined for points other than x=0 as
> >
> === 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
> <mailto:sanjivadayal%40yahoo.com>
> Personal email:sanjivadayal@hotmail.com
> <mailto:sanjivadayal%40hotmail.com>
>
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