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> 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.
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>
> ________________________________
>
> From: Math4u@yahoogroups.com
> [mailto:Math4u@yahoogroups.com] On Behalf
> Of Brian E. Jensen
> Sent: 01 October 2007 17:38
> To: Math4u@yahoogroups.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>
> To: Math4u@yahoogroups.com
> <mailto:Math4u@yahoogroups.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
Personal email:sanjivadayal@hotmail.com
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