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@getnet.net> 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@yahoo.com
Personal email:sanjivadayal@hotmail.com
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