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clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

Simple question on using contrapositive as a proof.

Question from book:
"Prove that if F is continuous on [a,b] and differentiable on (a,b) and if F'(x) is not zero for any x strictly between a and b, then

.

My question:
So I know that I have to prove this using the contrapositive. I also know that for (if p then q) the contrapositive is (if not q then not p).  So for my question I want to assume if F(b)=f(a) then ____________________________.

I don't know what to fill the ______ in with though?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Simple question on using contrapositive as a proof.

The questionn itself is the contrapositive of Rolle’s theorem – so if you take its contrapositive, you should get Rolle’s theorem. That’s just about it, I think.

clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

Re: Simple question on using contrapositive as a proof.

Really? I guess I'm not seeing it.
Rolle's says, "Let f be a function that is continuous on [a,b] and differentiable on (a,b) AND for which f(a)=f(b)=0. There exists at least one c, a<c<b for which f'(c)=0."

So i see that I'm assuming f(a)=f(b) but they dont necessarily have to equal 0?
This proof is killing me.

I also kind of feel like using the Intermediate Value Theorem... but that's not in this section of the book.
( what is: Generalized Mean Value Theorem, Cauchy's Remainder Theorem, L'Hospital's Rule
and Darboux's Theorem.)

clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

Re: Simple question on using contrapositive as a proof.

Could I possible say...

The mean value theorem states "Given a function f that is differentiable at all points strictly between a and x and continuous at all points on the closed interval from a to x, there
exists a real number c strictly between a and x such that [f(x)-f(a)]/x-a = f'(c)."

Thus we can prove the question by the contrapositive. Assume f(b)=f(a) and show that there must be an f'(x)=0. By the mean value theorem we have:
[f(b)-f(a)]/b-a = 0/(b-a) = 0 = f'(x) for x strictly between a and b.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Simple question on using contrapositive as a proof.

Historically, you shouldn't be using either.

Rolle's theorem typically isn't stated that f(a) = f(b) = 0, but f(a) = f(b)

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"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..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Simple question on using contrapositive as a proof.

Rolle's says, "Let f be a function that is continuous on [a,b] and differentiable on (a,b) AND for which f(a)=f(b)=0. There exists at least one c, a<c<b for which f'(c)=0."

In that case, use

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