Math Is Fun Forum

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Real Analysis Help

Hey guys. We are working on functions: increasing, decreasing, monotone, bounded, minimums, and maximums.

Last edited by sumpm1 (2009-03-07 02:37:23)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Real Analysis Help

Please do 1(a) and 1(b) yourself. They are straightforward enough.

I will help you with part of 1(c) because that one might be tricky.

Now do the same for

.

Last edited by JaneFairfax (2009-03-08 01:17:40)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Real Analysis Help

Jane, you need to stop this.  You don't yell at a person for asking a question, even for something they should know.  If you do that, people stop asking all questions, including the good ones.  Instead, you need to say something along the lines of, "What are you having problems with?"  I really don't want to have to say this again, please tone it down.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

ksmathwiz

Member

Registered: 2009-03-06

Posts: 4

Re: Real Analysis Help

Just use the definitions of the two that you provided. 

For part c I'll give you a hint that;

sinh(2x) = 2sinh(x)cosh(x)
cosh(2x) = cosh^2(x) + sinh^2(x)

Again use the definition of cosh(x) and sinh(x) and manipulate the numbers to get their identities.

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Re: Real Analysis Help

Hey guys, thanks for the help.

@ksmathwiz: Are you sure that the identity for cosh(2x) is not cosh(2x) = cosh^2(x) - sinh^2(x) rather than cosh(2x) = cosh^2(x) + sinh^2(x)?

Thanks

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Real Analysis Help

ksmathwiz’s formulas are correct.

ksmathwiz

Member

Registered: 2009-03-06

Posts: 4

Re: Real Analysis Help

Here:     (e^2x + e^-2x )/2 
          = (2e^2x + 2e^-2x)/4
          = (e^2x + 2 + e^-2x + e^2x– 2 + e^-2x)/4

You finish it.

Last edited by ksmathwiz (2009-03-07 04:37:21)

Muggleton

Member

Registered: 2009-01-15

Posts: 65

Re: Real Analysis Help

We can prove that

is increasing by contradiction.

Suppse it is not. Then there exist

in I such that

.

Now

and

.

Also

is either

or

.

Thus we have either

or

. This contradicts the fact that both f and g are increasing on I.

Therefore

is increasing on I.

You can prove that

is also increasing on the interval I by the same method.

Last edited by Muggleton (2009-03-07 07:42:47)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Real Analysis Help

Incidentally: