Absolute Extrema on Open Intervals

[Qube]

Homework Statement

Problem 2:

Find the absolute extreme values of f(x) on the interval (1, infinity).

Homework Equations

1. We say that f(x) has an absolute (or global) maximum at if for every x in the domain we are working on.
2. We say that f(x) has a relative (or local) maximum at if for every x in some open interval around .
3. We say that f(x) has an absolute (or global) minimum at if for every x in the domain we are working on.
4. We say that f(x) has a relative (or local) minimum at if for every x in some open interval around .

The Attempt at a Solution

It seems as if this is an indiscretion by the teacher. Absolute extrema don't occur on open intervals except in the case of (-∞,∞) such as in case of sin(x). Am i missing something?

 

[Student100]

The Attempt at a Solution

It seems as if this is an indiscretion by the teacher.

That is not true. Look at the end behavior of the function.

Also, I'm confused of what you mean by "except in the case of $(-\infty, \infty)$" every unspecified domain is assumed to be on that interval, and checking for absolute extrema is the same.

The theorem you want says something like this: If f has an absolute extremum on an open interval (a, b) then it must occur at a critical point of f.

Then look at the limits:

$\lim_{x\to a^+}f(x) <br /> \lim_{x\to b^-}f(x)$

 

[vela]

If you have a closed interval for the domain, you can find absolute extrema, but on an open interval you're not guaranteed to find them. This is not the same, however, as saying that you will not find an absolute extremum on the open interval.

Take for example f: (-1,1)->R defined by f(x)=x². It has an absolute minimum at x=0, but no absolute maximum.