Lars Olsen proof of Darboux's Intermediate Value Theorem for Derivatives

• Dec 28, 2022

• #1

Hall

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Here is Lars Olsen's proof. I'm having difficulty in understanding why ##y## will lie between ##f_a (a)## and ##f_a(b)##. Initially, we assumed that ##f'(a) \lt y \lt f'(b)##, but ##f_a(b)## doesn't equal to ##f'(b)##.

 

Last edited by a moderator: Dec 29, 2022

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• Dec 28, 2022

• #2

malawi_glenn

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##y## will lie between ##f_a(a)## and ##f_b(b)##.
We also have by the construction of the functions ##f_a## and ##f_b## that ##f_a(b) = f_b(a)##.
Thus, ##y## will lie between ##f_a(a)## and ##f_a(b)## or between ##f_b(a)## and ##f_b(b)##.

What is it that you do not understand? Can you come up with a differentiable function ##f## for which the above is not true?

 

• Dec 28, 2022

• #3

Hall

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malawi_glenn said:

##y## will lie between ##f_a(a)## and ##f_b(b)##.
We also have by the construction of the functions ##f_a## and ##f_b## that ##f_a(b) = f_b(a)##.
Thus, ##y## will lie between ##f_a(a)## and ##f_a(b)## or between ##f_b(a)## and ##f_b(b)##.

What is it that you do not understand? Can you come up with a differentiable function ##f## for which the above is not true?

## f_a(a) = f'(a) \lt y \lt f'(b) = f_b(b)##, that is correct. But how ##f_a(b) = f'(b)##?

 

• Dec 28, 2022

• #4

malawi_glenn

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Hall said:

But how fa(b)=f′(b)?

where is that written/used?
In other words, where do you read ##f_a(b) = f_b(b)##?

 

• Dec 28, 2022

• #5

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malawi_glenn said:

where is that written/used?
In other words, where do you read ##f_a(b) = f_b(b)##?

I don't know, sir, but I cannot see simply how ##y## lies between ##f_a(a)## and ##f_a(b)##.

 

• Dec 28, 2022

• #6

Hall

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##f_a(b) = \frac{ f(a) - f(b)}{a-b}##, by Mean Value Theorem there is some ##c \in (a,b)## such that ##f_a(b) = f'(c)##. Why ##y## necessarily need to lie between ##f'(a) and f'(c)##? We assumed it to lie between ##f'(a)## and ##f'(b)##.

 

• Dec 28, 2022

• #7

malawi_glenn

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Hall said:

I don't know, sir, but I cannot see simply how ##y## lies between ##f_a(a)## and ##f_a(b)##.

You have two cases to consider.
As I wrote, which is in the article.:
We have by the construction of the functions ##f_a## and ##f_b## that ##f_a(b) = f_b(a)##.
Thus, ##y## will lie between ##f_a(a)## and ##f_a(b)## or between ##f_b(a)## and ##f_b(b)##.

Work it out with the function say ##f(x) = x^2## and let ##a=-1## and ##b=2## or something.

 

Last edited: Dec 28, 2022

• Dec 28, 2022

• #8

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.

If ##y## lies between ##f'(a)## and ##f'(b)##, then it is surely to between the end-points of one of the graphs.

It was not very obvious for me to see, I was considering ##f_a(a)## and ##f_a(b)## individually and not in connection with that "or" ##f_b(a)## and ##f_b(a)##.

 

• Dec 28, 2022

• #9

malawi_glenn

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Are you okay with the proof now?

 

• Dec 28, 2022

• #10

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malawi_glenn said:

Are you okay with the proof now?

Yes.

 

What is the Darboux's Intermediate Value Theorem for Derivatives?

The Darboux's Intermediate Value Theorem for Derivatives states that if a function is differentiable on an interval and takes on two different values at the endpoints of the interval, then it must also take on every value in between those two values at some point within the interval.

Who is Lars Olsen?

Lars Olsen is a mathematician who is known for his proof of the Darboux's Intermediate Value Theorem for Derivatives. He was a Danish mathematician who lived from 1861 to 1919.

What is the significance of Lars Olsen's proof of the Darboux's Intermediate Value Theorem for Derivatives?

Lars Olsen's proof is significant because it provided a rigorous and complete proof of the theorem, which was previously only proven by intuition. His proof also helped to solidify the importance of the theorem in calculus and analysis.

What are the key steps in Lars Olsen's proof of the Darboux's Intermediate Value Theorem for Derivatives?

The key steps in Lars Olsen's proof include using the Mean Value Theorem to show that the derivative of the function must take on all values between the two endpoint values, and then using induction to show that the function itself must also take on all values between the two endpoint values.

How does Lars Olsen's proof of the Darboux's Intermediate Value Theorem for Derivatives relate to other proofs of the theorem?

Lars Olsen's proof is considered one of the most elegant and complete proofs of the Darboux's Intermediate Value Theorem for Derivatives. It is also closely related to other proofs of the theorem, such as the Brouncker's proof and the Bolzano's proof, which also use the Mean Value Theorem as a key step.