The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and *DIFFERENTIABLE ON THE INTERVAL (a,b)*, then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b].

In the following example, we'll see how this fails.

Let f(x) = (3/4) * (x - 4)^(4/3) + 1

If we consider the derivative of f'(x) = (x - 4)^(1 / 3), that is, f''(x) = 1 / (3(x - 4)^(2 / 3)), we have a vertical asymptote at x=4 and the mean value theorem doesn't care about it even though both sides increase indefinitely. We can find the area under the curve of f''(x) by considering an interval that includes x=4. That is, there is NO derivative f''(4) - sorry to disappoint you but infinity is not a derivative.

*SO TAKE NOTE: f ' (x) does NOT HAVE a derivative at x=4*

*This fact goes contrary to the statement of the mean value theorem in the mainstream which states that a function must be differentiable over the given interval in order to apply the mean value theorem, that is, f''(4) is not defined*

I first revealed these facts to the mainstream community as far back as 2005. I was called also sorts of names and libeled in every possible way by the scum who call themselves mainstream mathematics academics.

Till this day they refuse to be corrected because for them sound knowledge and truth is not important, only their garbage known as set theory and topology is important to them because it allows them to bullshit as much as they please or so they imagine. After all, you can back up garbage theory with ill-formed concepts to no end. You have the infinitely stupid morons like David Hilbert (far worse than his crank "father" Cantor whom he idolised) and the bogus ZFC foundations of modern mythmatics.

I am a voice crying out in the wilderness of mainstream mathematics academia. They are all blind fools being led about by myths and subscribing to the cult of mainstream mythmatics.

Stupid can't be corrected. Add arrogance, incompetence, ignorance, intransigence, unbelievable stupidity and jealousy to the very toxic attitude of the Church of Academia and what you have is a very stinky cologne.

One would think that given the importance of the mean value theorem, mainstream mathematics academics would have at least tried to get it right. But how can they if they've never understood its significance? That is, the mean value theorem *IS* the *FUNDAMENTAL THEOREM OF CALCULUS*

The *CORRECT AND REMARKABLE STATEMENT* of the mean value theorem follows.

*Given any smooth function f ' (x), the arithmetic mean (more accurate name is level term) of all the y-ordinates of f ' over the interval [a,b] is given by: f'(c) = [f(b)-f(a)]/(b-a) where a ≤ c ≤ b*

That such a c exists, is an unremarkable property of smooth functions, that is, there is at least one
y–ordinate which represents the level y–ordinate of ALL (innumerably many) y–ordinates in the interval [a,b].

Britannica:
"The theorem states that the slope of a line (secant line) connecting any two points on a “smooth” curve is the same as the slope of some line tangent to the curve at a point between the two points. In other words, at some point the slope of the curve must equal its average slope."

There is nothing remarkable about such a secant line as there are innumerably many such secant lines that have the same slope as that of some tangent to curve. NOTE AGAIN that this definition FAILS if there is a tangent line to a given curve such as f(x)=x^3 for which idiotic mainstream academics claim there is a tangent line at x=0. Mainstream academics claim that x^3 is differentiable at x=0, but there is no f'(0)=[f(b)-f(a)]/(b-a) and YET ACCORDING to the mean value theorem there should be! My 100% rigorous New Calculus DOES NOT allow tangent lines at points of inflection.

*LATE EDIT:*

Some dimwit wrote me and said that the mean value theorem has nothing to do with second derivatives. Tsk, tsk. The world is full of stupid people. Simply replace f'(x) with f(x) and f ''(x) with f'(x).

Fine.

Example: Let f(x) = (x - 4)^(1 / 3) so that f '(x) = 1 / (3(x - 4)^(2 / 3)). (**)

The Mean Value Theorem states that if a function f(x) = (x - 4)^(1 / 3) is continuous on the closed interval [a,b] and *DIFFERENTIABLE ON THE INTERVAL (a,b)*, then there exists a point c in the interval (a,b) such that 1 / (3(c - 4)^(2 / 3)) is equal to the function's average rate of change over [a,b].

In fact, there *exist TWO cees* in the example above over the interval [3,5] (**).

However, (x - 4)^(1 / 3) is *NOT differentiable* on any interval containing x=4 because there is *NO derivative at x=4,* ie. 1 / (3(4 - 4)^(2 / 3)) = 1/0 is undefined.

Hope you get it now!

How many crazy imbecilic hip hop humping fuckdogs of math should South Africa have?

It surely has John Gabriel.

So the population of South Africa is 60 million.

The population of Canada, where the crackpot Dan Christensen is from is 38 million, roughly 1/2 of South Africa.

Canada has the crackpot spammers Rick_s, Dan Christensen.

That portends the statistic that there are 3 more crazy lunatics in South Africa, yet to emerge and make a nuisance of themselves in sci.math & sci.physics.

God ever help us....

STUDENTS BEWARE: Don't be a victim of AP's (or JG's) fake math and science

On Thursday, July 21, 2022 at 3:47:22 AM UTC-4, Archimedes Plutonium wrote:
> How many crazy imbecilic...

When will you learn, Archie Poo???

AP is a malicious internet troll who wants only to mislead and confuse you. He may not be all there, but his fake math and science can only be meant to promote failure in schools. One can only guess at his motives.

In AP's OWN WORDS here:

“Primes do not exist, because the set they were borne from has no division.”
--June 29, 2020

“The last and largest finite number is 10^604.”
--June 3, 2015

“0 appears to be the last and largest finite number”
--June 9, 2015

“0/0 must be equal to 1.”
-- June 9, 2015

“0 is an infinite irrational number.”
--June 28, 2015

“No negative numbers exist.”
--December 22, 2018

“Rationals are not numbers.”
--May 18, 2019

According to AP's “chess board math,” an equilateral triangle is a right-triangle.
--December 11, 2019

Which could explain...

“The value of sin(45 degrees) = 1.”
--May 31, 2019

AP deliberately and repeatedly presented the truth table for OR as the truth table for AND:

“New Logic
AND
T & T = T
T & F = T
F & T = T
F & F = F”
--November 9, 2019

AP seeks aid of Russian agents to promote failure in schools:

"Please--Asking for help from Russia-- russian robots-- to create a new, true mathematics [sic]. What I like for the robots to do, is list every day, about 4 Colleges ( of the West) math dept, and ask why that math department is teaching false and fake math, and if unable to change to the correct true math, well, simply fire that math department until they can find professors who recognize truth in math from fakery...."
--November 9, 2017

And if that wasn't weird enough...

“The totality, everything that there is [the universe], is only 1 atom of plutonium [Pu]. There is nothing outside or beyond this one atom of plutonium.”
--April 4, 1994

“The Universe itself is one gigantic big atom.”
--November 14, 2019

AP's sinister Atom God Cult of Failure???

“Since God-Pu is marching on.
Glory! Glory! Atom Plutonium!
Its truth is marching on.
It has sounded forth the trumpet that shall never call retreat;
It is sifting out the hearts of people before its judgment seat;
Oh, be swift, my soul, to answer it; be jubilant, my feet!
Our God-Pu is marching on.”
--December 15, 2018 (Note: Pu is the atomic symbol for plutonium)

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

On 7/20/2022 9:13 PM, Ιωάννης Γαβριήλ wrote:

lern engilsh idiette

> The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and *DIFFERENTIABLE ON THE INTERVAL (a,b)*,

Shut up idiot.

On 21/07/2022 05:13, Ιωάννης Γαβριήλ wrote:
> The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and *DIFFERENTIABLE ON THE INTERVAL (a,b)*, then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b].

Yes, for MVT to apply, f' (FIRST derivative of f) must exist on open interval (a,b). There's no
requirement for f'' to exist.

>
> In the following example, we'll see how this fails.
>
> Let f(x) = (3/4) * (x - 4)^(4/3) + 1
>
> If we consider the derivative of f'(x) = (x - 4)^(1 / 3), that is, f''(x) = 1 / (3(x - 4)^(2 / 3)), we have a vertical asymptote at x=4 and the mean value theorem doesn't care about it even though both sides increase indefinitely. We can find the area under the curve of f''(x) by considering an interval that includes x=4. That is, there is NO derivative f''(4) - sorry to disappoint you but infinity is not a derivative.
>
> *SO TAKE NOTE: f ' (x) does NOT HAVE a derivative at x=4*

But f' exists on (a,b), and f is continuous on [a,b]. [BTW, what are you taking for a,b?]

MVT applies here.

>
> *This fact goes contrary to the statement of the mean value theorem in the mainstream which states that a function must be differentiable over the given interval in order to apply the mean value theorem, that is, f''(4) is not defined*

4''(4) is not defined, but that doesn't matter. MVT still applies - it does not require 2nd (or
higher) derivatives of f to exist.

Are you claiming the MVT conclusion is false, and f is some sort of counter-example? What are your
a, b?

>
> I first revealed these facts to the mainstream community as far back as 2005. I was called also sorts of names and libeled in every possible way by the scum who call themselves mainstream mathematics academics.

Well, you do sound quite confused! :) MVT applies to your f with no problem... You seem to be
claiming some sort of problem...

>
> Till this day they refuse to be corrected because for them sound knowledge and truth is not important, only their garbage known as set theory and topology is important to them because it allows them to bullshit as much as they please or so they imagine. After all, you can back up garbage theory with ill-formed concepts to no end. You have the infinitely stupid morons like David Hilbert (far worse than his crank "father" Cantor whom he idolised) and the bogus ZFC foundations of modern mythmatics.
>
> I am a voice crying out in the wilderness of mainstream mathematics academia. They are all blind fools being led about by myths and subscribing to the cult of mainstream mythmatics.

OK, now you're coming across as a regular crank.
Mike.

Gabriel pretending to be Leonard Nimoy of Star Trek in his logo. What does Dan Christensen logo remind one of??? A ulcerated arsehole???
> How many crazy imbecilic hip hop humping fuckdogs of math should South Africa have?
>
> It surely has John Gabriel.
>
> So the population of South Africa is 60 million.
>
> The population of Canada, where the crackpot Dan Christensen is from is 38 million, roughly 1/2 of South Africa.
>
> Canada has the crackpot spammers Rick_s, Dan Christensen.
>
> That portends the statistic that there are 3 more crazy lunatics in South Africa, yet to emerge and make a nuisance of themselves in sci.math & sci.physics.
>
> God ever help us....