Understanding Corollary 6.2.2 of B&S Theorem 6.2.1

[Math Amateur]

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...

I am focused on Chapter 6: Differentiation ...

I need help in fully understanding the corollary to Theorem 6.2.1 ...Theorem 6.2.1 and its corollary ... ... read as follows:
View attachment 7295
Can someone please demonstrate how to prove Corollary 6.2.2 above ...Note that ... obviously ... if the derivative of \(\displaystyle f\) exists then since \(\displaystyle c\) is an interior point at which \(\displaystyle f\) has a relative extremum then \(\displaystyle f'(c) = 0\) by Theorem 6.2.1 ... maybe this forms part of the proof ...

Hope someone can help ...

Peter

 

[GJA]

Hi Peter,

Either the derivative of $f$ at $x=c$ exists or it doesn't, and these are the only two possibilities. If it does, then $f'(c)=0$ from the theorem.

 

[Math Amateur]

Hi Peter,

Either the derivative of $f$ at $x=c$ exists or it doesn't, and these are the only two possibilities. If it does, then $f'(c)=0$ from the theorem.

Thanks GJA ...

Just another question ... why does f need to be continuous ...?

Peter

 

[GJA]

Hi Peter,

I don't see the need for assuming continuity. Just like the theorem statement, all you really need is the knowledge that $f$ has a local extremum at $x=c$.

 

[Math Amateur]

Hi Peter,

I don't see the need for assuming continuity. Just like the theorem statement, all you really need is the knowledge that $f$ has a local extremum at $x=c$.

Thanks GJA ...

Now ... just to be very clear ... you are saying that Bartle and Sherbert have put an irrelevant/unnecessary assumption in the corollary to the Interior Extremum Theorem ... Is that correct?

Peter*** EDIT ***

To add to my perplexity I have just discovered a similar corollary to the Interior Extremum Theorem in Manfred Stoll's book "Introduction to Real Analysis ... in the corollary, Stoll characterizes f as continuous on a closed interval ... whereas Bartle and Sherbert define f as continuous on an interval I ... ...

The theorem and its corollary in Stoll read as follows:View attachment 7297Since continuity is stated as an assumption in both Bartle and Sherbert and Stoll, it seems we should at least take it seriously ... are you sure your proof (that does not appear to need continuity) is valid ...

Note that I have to confess that I cannot fault your proof ...!?

 

[GJA]

Hi Peter,

I'm saying I don't see the value of the continuity assumption based on what you've shared. Perhaps the authors had a specific reason for including it, and, if they did, it's unclear to me what that purpose was.

Edit: I could speculate a reason or two, but as far as "proving" the corollary it appears superficial because the theorem statement is stated in a more general form (and thus subsumes the corollary).

 

[GJA]

Hi Peter,

Since Stoll uses a closed bounded (i.e., compact) interval, a global maximum and minimum (which are a fortiori local maximums and minimums, respectively) are guaranteed to exist since $f$ is assumed continuous. Bartle and Sherbert don't specify an interval type for $I$, so they must assume the existence of a local extremum (e.g., $f(x) = x$ has no local extrema on $(0,1)$).

If I had to speculate as to why both sets of authors choose to include the continuity assumption, I would guess that they are attempting to provide the reader with a tool (the corollary) that finds wide use in applications/examples. What do I mean here? Suppose you are working on a mathematically formalized problem that involves a continuous function on a compact (i.e., closed and bounded) interval $[a,b]$. Now in some cases simply knowing that this function possesses a global maximum and minimum on $[a,b]$ is all you need to know.

Sometimes, however, you may want to know the precise locations (i.e. the "exact" values for $x$ - exact in quotes because in practice one usually must numerically approximate the values) for the local extrema of $f$ on $[a,b]$ (e.g., perhaps $f$ represents the force exerted on a piece of wire, which is modeled by the interval $[a,b]$. Then one may be interested in knowing the point on the wire that experiences the greatest force). In this case, the theorem that guarantees the existence of these extrema is of little to no value to you, as there is nothing in the proof of that theorem that tells you how to actually locate the extrema you know to exist. This is where the corollary comes in. It says that if you need to know the "exact" location of the function's extrema, then you should find the values of $x$ where $f'(x)$ is zero and fails to exist. One must take care, however, to note that not every point one finds in this manner corresponds to a local extremum (i.e., the converse of the corollary is not true: e.g., $f(x)=\sqrt[3]{x}$ on $[-1,1]$ does not have a local extremum at $x=0$, even though $f'(0)$ DNE; similarly, $f(x)=x^{3}$ does not possesses a local extremum at $x=0$, even though $f'(0)=0$). Nevertheless, the values you determine by using this process are the candidates for the extrema you choose to hold dear on the problem you're solving. Stoll's remark after the corollary tries to warn you what can happen if the desired extrema sits at an endpoint of $[a,b]$ instead of interior to it (as is assumed in the corollary's statement). I must reiterate, though, that this is only my guess as to why they are formulating things this way.

Note that I have to confess that I cannot fault your proof...!?

I assure you that your curiosity and genuine desire to understand are never met with even the slightest indignation.

 

[Math Amateur]

Hi Peter,

Since Stoll uses a closed bounded (i.e., compact) interval, a global maximum and minimum (which are a fortiori local maximums and minimums, respectively) are guaranteed to exist since $f$ is assumed continuous. Bartle and Sherbert don't specify an interval type for $I$, so they must assume the existence of a local extremum (e.g., $f(x) = x$ has no local extrema on $(0,1)$).

If I had to speculate as to why both sets of authors choose to include the continuity assumption, I would guess that they are attempting to provide the reader with a tool (the corollary) that finds wide use in applications/examples. What do I mean here? Suppose you are working on a mathematically formalized problem that involves a continuous function on a compact (i.e., closed and bounded) interval $[a,b]$. Now in some cases simply knowing that this function possesses a global maximum and minimum on $[a,b]$ is all you need to know.

Sometimes, however, you may want to know the precise locations (i.e. the "exact" values for $x$ - exact in quotes because in practice one usually must numerically approximate the values) for the local extrema of $f$ on $[a,b]$ (e.g., perhaps $f$ represents the force exerted on a piece of wire, which is modeled by the interval $[a,b]$. Then one may be interested in knowing the point on the wire that experiences the greatest force). In this case, the theorem that guarantees the existence of these extrema is of little to no value to you, as there is nothing in the proof of that theorem that tells you how to actually locate the extrema you know to exist. This is where the corollary comes in. It says that if you need to know the "exact" location of the function's extrema, then you should find the values of $x$ where $f'(x)$ is zero and fails to exist. One must take care, however, to note that not every point one finds in this manner corresponds to a local extremum (i.e., the converse of the corollary is not true: e.g., $f(x)=\sqrt[3]{x}$ on $[-1,1]$ does not have a local extremum at $x=0$, even though $f'(0)$ DNE; similarly, $f(x)=x^{3}$ does not possesses a local extremum at $x=0$, even though $f'(0)=0$). Nevertheless, the values you determine by using this process are the candidates for the extrema you choose to hold dear on the problem you're solving. Stoll's remark after the corollary tries to warn you what can happen if the desired extrema sits at an endpoint of $[a,b]$ instead of interior to it (as is assumed in the corollary's statement). I must reiterate, though, that this is only my guess as to why they are formulating things this way.
I assure you that your curiosity and genuine desire to understand are never met with even the slightest indignation.

Thanks GJA ...

I really appreciate your helpful and thoughtful analysis ...

Thanks again,

Peter

 

[solakis1]

Thanks GJA ...

Just another question ... why does f need to be continuous ...?

Peter

Suppose that the word continuous is not mentioned ,and let us examine whether f is continuous at c or not.

Case I : f is not continuous at c. But we know that differentiability implies continuity and by contrapositive law : not continuity implies not differentiability.

Hence the derivative at c does not exist

Case II : f is continuous at c,then by the law of excleded middle the derivative at c exists or does not exist.

If it exists by the previous theorem : f'(c)=0

If it does not exist there is nothing to be examined.

Hence whether the function is continuous or not ,by using the law proof by cases of logic the derivative at c either does not exist or f'(c)=0

Hence if the Author does not mention the word continuous in his hypothesis the result is the same with the case where the Author mention the word continuous in his hypothesis. .

However the word continuous in the hypothesis does not participate in the 100% correct proof given by GJA