# Show existence of a,b

#### evinda

##### Well-known member
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Hello!!! Let $I$ an interval and $f: I \to \mathbb{R}$ a differentiable (as many times as we want) function.
If $\xi \in I$ with $f''(\xi) \neq 0$, then there are $a,b \in I: a< \xi <b$ and $\frac{f(b)-f(a)}{b-a}=f'(\xi)$.

Hint: First suppose that $f'(\xi)=0$ and show that at $\xi$ we have a local extremum. Then find (for example with the intermediate value theorem) $a,b$ with $a< \xi<b$ and $f(a)=f(b)$, therefore... etc.
For $f'(\xi) \neq 0$, consider the function $g(x)=f(x)-f'(\xi)x$ to get reduced to the previous case.

So we suppose that $f'(\xi)=0$.

How do we show that $f$ has a local extremum at $\xi$ ?

I have thought to calculate the Taylor series of second order.

Then, $f(x)=f(\xi)+f'(\xi) (x-\xi)+\frac{f''(\xi)}{2!}(x-\xi)^2=f(\xi)+\frac{f''(\xi)}{2}(x-\xi)^2$.

From this we get that $f(x)-f(\xi) \neq 0$.

But this does not help somehow, does it?

How else can we start the proof? #### Klaas van Aarsen

##### MHB Seeker
Staff member
Hello!!!

Let $I$ an interval and $f: I \to \mathbb{R}$ a differentiable (as many times as we want) function.
If $\xi \in I$ with $f''(\xi) \neq 0$, then there are $a,b \in I: a< \xi <b$ and $\frac{f(b)-f(a)}{b-a}=f'(\xi)$.

Hint: First suppose that $f'(\xi)=0$ and show that at $\xi$ we have a local extremum. Then find (for example with the intermediate value theorem) $a,b$ with $a< \xi<b$ and $f(a)=f(b)$, therefore... etc.
For $f'(\xi) \neq 0$, consider the function $g(x)=f(x)-f'(\xi)x$ to get reduced to the previous case.

So we suppose that $f'(\xi)=0$.

How do we show that $f$ has a local extremum at $\xi$ ?
Hey evinda !!

If $f'(\xi)=0$ then we either have a local extremum or a saddle point don't we?
Can it be a saddle point? I have thought to calculate the Taylor series of second order.

Then, $f(x)=f(\xi)+f'(\xi) (x-\xi)+\frac{f''(\xi)}{2!}(x-\xi)^2=f(\xi)+\frac{f''(\xi)}{2}(x-\xi)^2$.

From this we get that $f(x)-f(\xi) \neq 0$.
Couldn't we still have that $f(x)-f(\xi) = 0$?

Suppose we pick $I=[-2,2]$, $f(x)=x^2-1$, $x=-1$, and $\xi=+1$.
Don't we have $f(x)-f(\xi)=0$ then? #### evinda

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Hey evinda !!

If $f'(\xi)=0$ then we either have a local extremum or a saddle point don't we?
Can it be a saddle point? When do we have a saddle point given a single-valued function?  Couldn't we still have that $f(x)-f(\xi) = 0$?

Suppose we pick $I=[-2,2]$, $f(x)=x^2-1$, $x=-1$, and $\xi=+1$.
Don't we have $f(x)-f(\xi)=0$ then? Oh yes, right!!!  #### Klaas van Aarsen

##### MHB Seeker
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When do we have a saddle point given a single-valued function?
Ah, sorry, it's not called a saddle point for a function of one variable. Instead it's called an inflection point.
Such as is the case for $f(x)=x^3$ at $x=0$.
It has $f'(0)=0$ doesn't it? But it's not a local extremum is it? #### evinda

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MHB Site Helper
Ah, sorry, it's not called a saddle point for a function of one variable. Instead it's called an inflection point.
Such as is the case for $f(x)=x^3$ at $x=0$.
It has $f'(0)=0$ doesn't it? But it's not a local extremum is it? Ah, I see... A necessary condition for $x$ to be an inflection point is that $f''(x)=0$.

So in our case since $f'(\xi)=0$ and $f''(\xi) \neq 0$, we have that $\xi$ is a local extremum.

Then let $a, b \in \mathbb{R}$ with $a<b$ such that $a< \xi<b$ and $[a,b] \subseteq I$. Then $f$ is continuous on $[a,b]$.
From the Intermediate Value Theorem we have that for every $y_0$ between $f(a)$ and $f(b)$, there exists a number $x_0 \in [a,b]$ such that $f(x_0)=y_0$.

First of all can we pick $a,b$ such that $a< \xi<b$ ? Because here $\xi$ is specific.

Secondly, the condition $f(x_0)=y_0$ does not help to conclude that $f(a)=f(b)$, does it? #### Klaas van Aarsen

##### MHB Seeker
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Ah, I see... A necessary condition for $x$ to be an inflection point is that $f''(x)=0$.

So in our case since $f'(\xi)=0$ and $f''(\xi) \neq 0$, we have that $\xi$ is a local extremum.
Yep. Then let $a, b \in \mathbb{R}$ with $a<b$ such that $a< \xi<b$ and $[a,b] \subseteq I$. Then $f$ is continuous on $[a,b]$.
From the Intermediate Value Theorem we have that for every $y_0$ between $f(a)$ and $f(b)$, there exists a number $x_0 \in [a,b]$ such that $f(x_0)=y_0$.

First of all can we pick $a,b$ such that $a< \xi<b$ ? Because here $\xi$ is specific.

Secondly, the condition $f(x_0)=y_0$ does not help to conclude that $f(a)=f(b)$, does it?
Sure.
Assuming that $\xi$ is not on the boundary of $I$ (which does not seem to be given, but we do need that), we can always find $a,b\in I$ such that $a<\xi<b$. We can't be sure that for such $a,b$ we will have $f(a)=f(b)$ though. So let's pick $a_1,b_1\in I$ instead such that $a_1<\xi<b_1$.
Let's assume that $f$ has a local maximum at $\xi$ and $f(a_1)<f(b_1)$ for now.
Then we must be able to reach any intermediate value between $f(a_1)$ and $f(\xi)$ mustn't we? #### evinda

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MHB Site Helper
Sure.
Assuming that $\xi$ is not on the boundary of $I$ (which does not seem to be given, but we do need that), we can always find $a,b\in I$ such that $a<\xi<b$. We can't be sure that for such $a,b$ we will have $f(a)=f(b)$ though. So let's pick $a_1,b_1\in I$ instead such that $a_1<\xi<b_1$.
Let's assume that $f$ has a local maximum at $\xi$ and $f(a_1)<f(b_1)$ for now.
Then we must be able to reach any intermediate value between $f(a_1)$ and $f(\xi)$ mustn't we? Yes, since $f$ is continuous on $[a_1, \xi]$, we have from the intermediate value theorem that for evey $y_0$ between $f(a_1)$ and $f(\xi)$ there exists a number $x_0 \in [a_1, \xi]$ such that $f(x_0)=y_0$.

But how does this help? #### Klaas van Aarsen

##### MHB Seeker
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Yes, since $f$ is continuous on $[a_1, \xi]$, we have from the intermediate value theorem that for evey $y_0$ between $f(a_1)$ and $f(\xi)$ there exists a number $x_0 \in [a_1, \xi]$ such that $f(x_0)=y_0$.

But how does this help?
Doesn't that mean that there is an $a_2\in[a_1, \xi]$ such that $f(a_2)=f(b_1)$? #### evinda

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Doesn't that mean that there is an $a_2\in[a_1, \xi]$ such that $f(a_2)=f(b_1)$? Ah yes, right! Thus we have that $\frac{f(b_1)-f(a_2)}{b_1-a_2}=0=f'(\xi)$, i.e. we have found the desired $a,b$, right? Then, we suppose that $f'(\xi) \neq 0$. We consider the function $g(x)=f(x)-f'(\xi)x$.

We have that $g'(x)=f'(x)-f'(\xi)$.

So, $g'(\xi)=0$.

Furthermore, $g''(x)=f''(x)$ and so $g''(\xi) \neq 0$.

So from the previous case we have that there are $a, b \in I$ such that $\frac{g(b)-g(a)}{b-a}=g'(\xi)$.

From this we get that $\frac{f(b)-f'(\xi)b-f(a)+f'(\xi)b}{b-a}=g'(\xi)=0$.

But from this we get that $\frac{f(b)-f(a)}{b-a}=0$.

Have I done something wrong? #### Klaas van Aarsen

##### MHB Seeker
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Ah yes, right! Thus we have that $\frac{f(b_1)-f(a_2)}{b_1-a_2}=0=f'(\xi)$, i.e. we have found the desired $a,b$, right?
Yep. Then, we suppose that $f'(\xi) \neq 0$. We consider the function $g(x)=f(x)-f'(\xi)x$.

We have that $g'(x)=f'(x)-f'(\xi)$.

So, $g'(\xi)=0$.

Furthermore, $g''(x)=f''(x)$ and so $g''(\xi) \neq 0$.

So from the previous case we have that there are $a, b \in I$ such that $\frac{g(b)-g(a)}{b-a}=g'(\xi)$.

From this we get that $\frac{f(b)-f'(\xi)b-f(a)+f'(\xi)b}{b-a}=g'(\xi)=0$.

But from this we get that $\frac{f(b)-f(a)}{b-a}=0$.

Have I done something wrong?
Shouldn't it be $\frac{f(b)-f'(\xi)b-f(a)+f'(\xi)\,{\color{red}a}}{b-a}=g'(\xi)=0$? #### evinda

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Yep. Shouldn't it be $\frac{f(b)-f'(\xi)b-f(a)+f'(\xi)\,{\color{red}a}}{b-a}=g'(\xi)=0$? Ah I see, then we get the desired result!!! Do we have to look also at the case when $f$ has a local minimum at $\xi$ ? #### Klaas van Aarsen

##### MHB Seeker
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Ah I see, then we get the desired result!!!

Do we have to look also at the case when $f$ has a local minimum at $\xi$ ?
Strictly speaking, yes.
And also the case that $f(a_1)\ge f(b_1)$. I would usually hand wave it away saying that we can show the same result in those cases in similar fashion. We can can't we? 