Rolle’s Theorem Proof

In this video, I prove Rolle’s theorem, which says that if f(a) = f(b), then there is a point c between a and b such that f’(c) = 0. This theorem is quintessential in proving the mean-value theorem in Calculus. Along the way I prove Fermat’s theorem, which says that if f has a maximum/minimum at a point c, then f’(c) = 0

[michaelz]

Dr Peyam proves that English is a tonal language.

[connorhorman]

This is amazing, as I have a differential calculus final later today, and we learned about Rolle's Theorem, but I always wondered the proof for it.

[danielescotece]

This is trivial

-This post was made by Lagrange's GANG

[joeyjoey]

The proof really helped me out understanding the subtle concepts. You’re the best. But I still have two problems bothering me regarding rather minor issues first, @ 9:27 where you starts mentioning about inequality for the numerator f(c+h)-f(c)<=0, why does it include the equal sign? Is it because of the definition of global extreme values? And second for the denominator where you mentioned h>=0, why is there equal sign? As far as I know, shouldn’t it be h>0 according to the definition of limit as h is never zero because h is approaching 0 but never zero itself? I really appreciate your superb proof and I’ll find it truly grateful if you could answer my question. Have a great day!❤

[nathanisbored]

this is one of those things that feels so obvious it doesnt seem like it should need to be proved

[abhisheknandann]

Just want you to know your videos are helping an Indian college student. a lot. wish my college professors were like you. THANKS AGAIN, STAY SAFE.

[louighi]

I just love you man. Here in Germany you have such a good repute!

[erikkonstas]

2:31 And this is what mathematicians use everyday to stay fit.

[doodelay]

This seems super obvious when looking at a graph, but if u were doing or reading some function with no immediate access to it's graphical representation, this theorem would prove useful in that regard cause it'll help you intuit what the problem is saying

[pkvlogs]

well Dr. Peyam
wanna tell u about this discovery
mitchel rolle was quit curious about intermediate velocity of a particle tracing a undistorted path usually if someone ask then we gonna say oh yess its avg. speed was |X2 -X1 |/ t2 - t1 ok but what ab8 its velocity, speed while covering its journey then he had gone through basics of algebra and connect with the defination of velocity v(t) = X'(t) , t €(t1, t2)

0< v(t) < |x2 - x1| / a (t2 -t1)

0< a<1
clearly says v(t) must have its values equal to |X2 - X1| / (t2 - t1) as 0<a<1

if x2 = x1 then then we must v(t) =0 for t€(t1, t2)

a common example we can see in simple projected objects, swinging pendulms..

coz sup., inf. were the later terminology introduced after Rolle so why to use sup., inf. and so on... coz i hv seen it many times that the same kinda proof is being served while comes to rolles theorem proof.

vdo was good 🌿

[maxsch.]

2:48
Hi Dr. Peyam :)

Btw great video

[saidfilali]

can you help me.

We consider (F) to be a numerical function.

- When the derivative of (F) is greater than or equal to 0, we say that the function (F) is increasing

- When the derivative of (F) is greater than 0 & not equal to 0, we say that the function (F) is strictly increasing

- When the derivative of (F) is less than or equal to 0, we say that the function (F) is decreasing

-When the derivative of (F) is less than 0 & not equal 0 , we say that the function (F) strictly decreasing

Our teacher said that there are exceptional cases where the derivative of (F) can be equal 0 at the ending points. than we say that the function is strictly increasing (or, strictly decreasing )

I really didn't get a good idea about this. So I want an example of this exceptional case ... I hope that you will make a video to explain it accurately ... Thank you and may God protect you, Dr. Peyam

[Karthik-lqgn]

Here is my proof (though not rigorous, I have never heard of Rolle's theorem before) :
There are two cases for the continuous function f where f(a) = f(b) and we'll proceed by showing the two cases are equivalent,


CASE 1: f'(a) is positive if f'(b) is negative or vice versa
CASE 2: f'(a) and f'(b) have same sign i.e, they are both positive or both negative. In this case it is quite obvious that there must exist some x = d where d < b such that f(a) = f(d) and f'(d) is negative if f'(a) is positive or vice versa. This is simply case 1.


Consider the points a, b again. It is possible to choose a and b such that they are arbitrarily close to each other and f(a) = f(b), i.e f(a) = f(b) and a - b can be made arbitrarily small and so if a - b ---> 0 then f'(a) - f'(b) ---> 0 (because, from a, if we go to the right the slope increases but from b if we go left the slope decreases) but if a and b can be made arbitrarily close to each then there must a exist a point c to which a and b are approaching to (since f is continuous), where f'(c) = f'(a) - f'(b) = 0 (where, c = a = b) and that concludes the proof. I am not going to watch the video until I get a reply...

[moawyahabdulrahman]

I'd like to see more of these proofs videos .

[isharauditha]

I hate math but I love to watch Dr Peyam's math videos

[wwebadgerse]

hey dr peyam! Im a 2nd year student and in my linear algebra course im trying to show that the infinite dimensional vector space R^infinity = {(x_1, x_2, ...) | x_i in R} has a basis which is not countable. Could you do something on these infinite dimensional vector spaces? i find them fascinating. thank you and i love your videos :)

[ricardoguzman]

This is great. Like the ham sandwich theorem. I prefer a ham sandwich on a rolle with butter, mayonnaise, and mustard, with a big stack of ham.

[ryanwickremasinghe]

Can a Maximum and a Minimum both exist within (a, b), and if so would it have any consequences?

[gordonchan]

Does Extreme Value theorem have to be proven as well?

[whythosenames]

Isn’t it just a special case of the theorem that there exists a c in R with f‘(c)=(f(b)-f(a))/(b-a)