1.
My function f(x) is a real function and the statement
is false.
Sanjiva
--- "Brian E. Jensen" <brianejensen@prodigy.net>
wrote:
> 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
> To: 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
>
> d abs(x)/dx = -1 for x<0; undefined for x=0; 1 for
> x>0 (8)
>
> As a result of the fact that computer algebra
> programs such as
> Mathematica
> generically deal with complex variables (i.e., the
> definition of
> derivative
> always means complex derivative), d abs(x)/dx
> correctly returns
> unevaluated
> by such software.
>
> Jim FitzSimons
>
> --- In Math4u@yahoogroups.com, sanjiva dayal
> <sanjivadayal@...>
> wrote:
> >
> > Hello,
> > 1.
> > Let me put my question more clearly:-
> > Whether the following statement is true or
> false:-
> > "f(x) is defined for all real x and f(x) is
> > differentiable for all real x and f(x) has two
> local
> > maximas then there must be a local minima
> between
> > these two local maximas".
> > 2.
> > The above statement is false. Can anyone give an
> > example which proves that the above statement is
> > false?
> > 3.
> > I have put this question because I find many
> such
> > false statements in books and being taught by
> > teachers.
> > Sanjiva
> > --- w7anf <cherry@...> wrote:
> >
> > > A friend said if a function is differentiable,
> then
> > > it
> > > continuous. My example is not continuous, so
> it is
> > > not
> > > differentiable. I will have to find another
> example.
> > > Jim FitzSimons
> > >
> > > --- In Math4u@yahoogroups.com, "w7anf"
> <cherry@>
> > > wrote:
> > > >
> > > > f'(x) = -4x^3 + 24x^2 -44x + 24
> > > > f'(1) = -4 + 24 -44 + 24 = 0
> > > > f'(3) = -4*27 + 24*9 -44*3 + 24
> > > > f'(3) = -108 + 216 -132 + 24 = 0
> > > > The domain of f(x)is x not equal 2
> > > > by definition.
> > > >
> > > > Jim FitzSimons
> > > >
> > > > --- In Math4u@yahoogroups.com, "Brian E.
> Jensen"
> > > <brianejensen@>
> > > > wrote:
> > > > >
> > > > > I agree with the second Jim that f(2)
> exists
> > > and it is a
> > > minimum.
> > > > > I agree with the original question that if
> a
> > > function is
> > > > differentiable, there must be a minimum
> between 2
> > > maxima. But I
> > > > think Jim FitzSimons is talented so maybe I
> am
>
=== 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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