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
> > missing something.
> > > > Going back to Jim FitzSimons message
> > > > f(x)=-x*(x-4)*(x^2-4*x+6), x not equal 2
> > > > How did Jim come up with such an example, why
> > is this factored
> > > and why did Jim say there are maxima at x=1 and
> > x=3?
> > > > f(x)= -x^4 + 8x^3 - 22x^2 + 24x
> > > > f'(x) = -4x^3 + 24x^2 -44x + 24
> > > > If I substitute 1 or 3 for x into the above
> > equation, I don't
> > > get zero so there is no maximum or minimum at x=1
> > or x=3
> > > > Regards, Brian Jensen
> > > > From: Jim
> > > > To: Math4u@yahoogroups.com
> > > > Sent: Sunday, September 23, 2007 11:43 AM
> > > > Subject: [Math4u] Re: Maxima & Minima
> > > > A different Jim disagreed with Jim FitzSimons:
> > > > >Unless I misread your function f(2)=8 so it
> > does exist and is
> > a
> > > minima.
> > > >
> > > > --- In Math4u@yahoogroups.com, "w7anf" <cherry@>
> > wrote:
> > > > >
> > > > > False, here is an example.
> > > > > f(x)=-x*(x-4)*(x^2-4*x+6), x not equal 2
> > > > > f(x) has maximas at x=1 and x=3.
> > > > > The minima should be at x=2, but the function
> > does not exist
> > > > > at x=2. f(x) is differentiable at x=2.
> > > > > Jim FitzSimons
> > > > >
> > > > >
> > > > > --- In Math4u@yahoogroups.com, "sanjivadayal"
> > <sanjivadayal@>
> > > wrote:
> > > > > >
> > > > > > Whether the following statement is true or
> > false?
> > > > > > "If f(x) is a differentiable function, then
> > there must be at
> > > least
> > > > > one
> > > > > > minima between two maximas".
> > > >
> > >
> >
> >
> >
>
>
> 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@...
> Personal email:sanjivadayal@...
>
>
>
>
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