Functions and their Extrema: Understanding Absolute Maxima and Minima

[nDever]

Hey guys,

I have learned that for a function f defined on an open interval I(a, b), an absolute extremum may not occur. Is this because the domain may take on every value greater than a or less than b without ever equaling a or b?

[EDIT]

Disregard the question above. I just realized. For a function such as f(x) = x², there is no absolute maximum because as x increases, f(x) increases indefinitely.

 

[Simon Bridge]

Check your ideas:
Does $$e^{-x^2}+x^2$$ ... have an absolute maximum at x=0?

 

[nDever]

Check your ideas:
Does $$e^{-x^2}+x^2$$ ... have an absolute maximum at x=0?

Wouldn't that be a minimum?

 

[Simon Bridge]

how do you figure?

 

[nDever]

how do you figure?

At 0, y=1 and for all other x less than or greater than zero, y is equal to or greater than 1. Correct?

 

[Simon Bridge]

It's pretty flat about x=0 isn't it? ... how about:
$$2e^{-x^2} + \frac{1}{2}x^2$$

 

[nDever]

It's pretty flat about x=0 isn't it? ... how about:
$$2e^{-x^2} + \frac{1}{2}x^2$$

For this, wouldn't the absolute minima be at the troughs around x= 1 and x= -1 and the absolute maximum doesn't exist?

 

[Simon Bridge]

There is a local maximum at x=0, but it ain't global.

If there is one "absolute" or global minima, then it is "unique"... here there is no unique global minimum. Well done.

Technically they are at: $x = \pm \sqrt{-\ln{(1/4)}} \approx \pm 1.1773$ but I'm not really looking for exact here, just understanding.

There is no new math here, it's just a matter of getting used to the common language.

 

[nDever]

How did you post the mathematics in the format that you did? Tags?

 

[Simon Bridge]

If you quote my message you'll see the tags.

The bb code is [tex] (pronounced tech - as in technology) - to put equations inline like I did use [itex]. Everything written inside those tags is LaTeX (lay-tech).
This is an academic standard way to mark-up documents for publication: really worth learning. The forum server has an engine which can turn LaTeX markup into a graphic... It makes writing out math very very simple and clear.

There are loads of tutorials etc around the web, and you can get a "Latex" program for any platform for you to use at home.
The only problem is, when you tell someone "I am really into latex." they look at you funny.

 

[nDever]

OK, so then allow me to do another example.

for $$2e^{-x^2} + \frac{1}{2}x$$,

there wouldn't be any global extrema. Zero would be a local maximum, and the the local minimum would depend on the open interval. Is this correct?

 

[Simon Bridge]

Extrema are all local to themselves, so it is a fair characterization without specifying the region. I checked and it turns out that you can have more than one global extrema ... which is why the term "unique" gets applied when there is only one.

In $f(x)=x(x-1)(x+1)$ there are two extrema, neither are global.

The fun thing is that the function can be anything ... eg. does the dirac delta function have any extrema?

 

[nDever]

Extrema are all local to themselves

So you mean that all extrema are local, but not all local extrema are global?

 

[Simon Bridge]

That would be right.

I use local and global for relative and absolute - I don't think it matters.
But it is useful to play around and explore to deepen your understanding.
It's usually more important to be able to express words in math than math in words though.

So now you can use the language, and you learned to use LaTeX.
That should be enough for now :)

 

[nDever]

But it is useful to play around and explore to deepen your understanding.
It's usually more important to be able to express words in math than math in words though.

I agree. I am better at understanding the concept and creating abstractions in my mind than I am at mathematical terminology.

Thank you for all of your help.