Question

If f(x) is continuous on (0,1), then it reaches both its absolute maximum and minimum somewhere inside (0,1). True or False

Answer 1

What does being continuous mean?
Could you have a function that is continuous on (0,1) that looks like this:

Answer 2

I know that f(x) is continuous on closed interval [0,1], but im not sure if its continuous on (0,1). Im not sure if the question is trying to trick me or not.

Answer 3

Are you given a function? Or is this just a general question?

Answer 4

just a true or false question

Answer 5

No, the extreme value theorem is only valid for compact sets. Since \((0, 1)\) is not compact the extreme value theorem does not hold.

Answer 6

We can provide a counterexample to show this. Consider \(f(x) = \frac{1}{x}\) on the interval \((0, 1)\). It is continuous on that interval but it certainly does not attain a maximum.

Answer 7

For more information see http://en.wikipedia.org/wiki/Extreme_value_theorem

Answer 8

i think we used the mean value theorem

Answer 9

Yes... but this question does not involve the mean value theorem. Read the statement of the extreme value theorem carefully.

Answer 10

If we are working with functions of the form \(f : \mathbb{R} \to \mathbb{R}\) then the extreme value theorem states that \(f\) is continuous and consider \([a, b]\) a closed and bounded interval then \(f\) attains a maximum and minimum value on that interval.

Answer 11

so can it be continuous on both closed and open interval?

Answer 12

No, that's the reason it doesn't work in the example you provided. Because its an open interval.

Answer 13

The extreme value theorem only applies to closed and bounded intervals.

Answer 14

oh okay I see now thanks.