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

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)##.
 
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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?
 
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)##?
 
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)##?
 
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)##.
 
##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)##.
 
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)##.

1672245477224.png


Work it out with the function say ##f(x) = x^2## and let ##a=-1## and ##b=2## or something.
 
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Darboux.jpg
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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)##.
 
Are you okay with the proof now?
 
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malawi_glenn said:
Are you okay with the proof now?
Yes.
 
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