A function with no max or min at an endpoint

[mcastillo356]

TL;DR

I've found an exercise, and need to revisit the concepts involved, first of all, and eventually solve it. I have some clues, but at the same time, a need to start from zero.

Hi, PF

Although a function cannot have extreme values anywhere other than at endpoints, critical points, and singular points, it need not have extreme values at such points. There is an example of how a function need not have extreme values at a critical point or a singular point in 9th edition of "Calculus: A Complete Course", pg 237, Figure 4.18. It is more difficult to draw the graph of a function whose domain has an endpoint at which the function fails to have an extreme value. See Exercise 49 at the end of this section for an example of such a function.
I would like to know why $x=0$ is an endpoint of such function, and why does it fail to have an extreme value, for the first instance. My aim is to completely understand and solve it.

Greetings, Love!

 

[mcastillo356]

Sorry, lack of statement of the exercise, and clues to solve it I have
Now have no time. Let's leave for tomorrow
Love

 

[FactChecker]

It's not hard to draw such a function. Graph the sine function. Pick an interval that encloses the maximum and the minimum but goes beyond them to points that are neither a max or min. $[0,\pi]$ will do.

 

[mcastillo356]

Fine. Think the example I will put is going to be much more complicated. At first glance.
See later

 

[FactChecker]

If you are talking about local maximums and minimums, it is harder to avoid those at the endpoints. But it can be done.

 

[mfb]

Copying the function would have been useful for users who don't have that book.

You can avoid a local minimum/maximum at an endpoint using oscillating functions for example. $f(x)=x \sin(\frac 1 x)$ for $x>0$, $f(x)=0$ is continuous but oscillates in the $\pm x$ "cone" infinitely often as you approach x=0. If the function doesn't need to be continuous there is even more freedom how to do that.
You simply define the interval to end at x=0. It's arbitrary which range to look at.

 

[Mark44]

There's always the Weierstrass Function (https://en.wikipedia.org/wiki/Weierstrass_function) that is continuous everywhere, but nowhere differentiable.

 

[mcastillo356]

There's always the Weierstrass Function (https://en.wikipedia.org/wiki/Weierstrass_function) that is continuous everywhere, but nowhere differentiable.

I've taken a glance. Very advanced to me

Copying the function would have been useful for users who don't have that book.

You can avoid a local minimum/maximum at an endpoint using oscillating functions for example. $f(x)=x \sin(\frac 1 x)$ for $x>0$, $f(x)=0$ is continuous but oscillates in the $\pm x$ "cone" infinitely often as you approach x=0. If the function doesn't need to be continuous there is even more freedom how to do that.
You simply define the interval to end at x=0. It's arbitrary which range to look at.

Thanks, this is fine, it is what I was going to post.
This is going to be hard work for me. This island I hope will once again support my lack of rigor and inconsistency. I'm also in touch with "Rincón Matemático", and some of you will have noticed that at my first post I made a question I should already have known the answer to. In my confusing paragraph I say:

I would like to know why $x=0$ is an endpoint of such function

This is something I should know from

https://www.physicsforums.com/threads/what-kind-of-definition-is-this.1008348/

Where I said I had understood the definition, and thus, the concept. Well, still wondering how to face:

-The definition at my previous post: I'm thinking now about personal classes.
-The exercise proposed here: I'm thinking about PF, the Spanish Forum, a personal teacher.

Well, too many words, just to apologize about my first post, and to introduce at last my wonders about this thread's direction. Still don't made the first step: write the question and contribution or effort to solve it. This is just a statement of intent, an outline of what i wish was the thread, and a plea: I need time to draft properly all I have in mind.

Thanks! Love