Re: Showing derivative >0

Quote:

Originally Posted by

**jsndacruz** This problem comes from Edward J. Gaughan's Introduction to Analysis.

*The Problem*
(4.3.19) Show that

has precisely one root on the closed interval

*Attempted Solution*

Let

. Since each component of

is differentiable and therefore continuous on the interval. Then by the Intermediate Value Theorem, since

and

,

such that

.

Since

, if

, then

,

is monotonic increasing and therefore

,

.

*The issue*

The derivative of

at a point

is the limit of

as

. Is there a way to show that this limit is strictly greater than zero? Is it sufficient to find the limit of

at

and then say that since

is monotonic increasing, that this would be a lower bound for the derivative of

?

Rolle's Theorem states that in order for a function to pass through the x axis twice, it needs to turn somewhere. Therefore, if there are two roots, , then there exists some such that .

Anyway, your function is , its derivative is .

It's well known that when , and it's also relatively easy to see that in that region as well.

Therefore, for all , which means the function does not turn, and therefore there can not possibly be any more than one root in that interval.

Re: Showing derivative >0

Thanks Prove It.

The problem is that I can't use the equation for ; I can only use its limit form. I was trying to show the derivative is greater than zero because then I could apply Rolle's Theorem and show that since the derivative was strictly greater than zero, the function could not "turn" to get another zero.

Re: Showing derivative >0

Quote:

Originally Posted by

**jsndacruz** Thanks Prove It.

The problem is that I can't use the equation for

; I can only use its limit form. I was trying to show the derivative is greater than zero because then I could apply Rolle's Theorem and show that since the derivative was strictly greater than zero, the function could not "turn" to get another zero.

That's ridiculous. Why would you not be able to use the equation for the derivative? Just some simple analysis of the equation is enough to show that the derivative is positive for all x in that region.

Re: Showing derivative >0

I know, but we're just getting started on formally defining differentiation. It's an introductory analysis course and we're only allowed to use theorems/identities that we can prove. I don't know how to prove that the derivative of is using the limit definition of differentiation, so I can't use that form.

Re: Showing derivative >0

Re: Showing derivative >0

Quote:

Originally Posted by

**Prove It** Well first of all, the derivative of

is NOT

, it's

. Anyway..

Twas but a typo.

For the rest of it .. thanks for such a thorough proof. I would have never been able to come up with that myself.