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

[Hall]

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 $f'(b)$.

 

[malawi_glenn]

$y$ 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?

 

[Hall]

$y$ 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 $f_a(b) = f'(b)$?

 

[malawi_glenn]

But how fa(b)=f′(b)?

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

 

[Hall]

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 $f_a(b)$.

 

[Hall]

$f_a(b) = \frac{ f(a) - f(b)}{a-b}$, 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 $f'(b)$.

 

[malawi_glenn]

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.

 

[Hall]

.

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 $f_b(a)$.

 

[malawi_glenn]

Are you okay with the proof now?

 

[Hall]

Are you okay with the proof now?

Yes.