Saturday, March 11, 2006

Rolle's Theorem

Rolle's Theorem is one of the basic principles of elementary calculus. It is used to proof the Taylor Series which is used to prove Euler's Formula.

In today's blog, I present a proof based on the maxima and minima property of a continous function.

If you need a review of derivatives, start here.

If you need a review of continuous functions or closed intervals, start here.

If you need a review of the maxima and minima properties of continuous functions over a closed interval, start here.

Theorem: Rolle's Theorem

If a function f(x) is continuous over a closed interval [a,b], differentiable over that interval, with f(a)=0 and f(b)=0, then there exists a point c [a,b] such that f'(c) =0.

Proof:

(1) Because the function f is continuous over the closed interval [a,b], then there exists a maximum value (see Lemma 3 here) and a minimum value (see the Theorem here)

(2) If f has any positive values, then let c = its maximum value over the closed interval [a,b]

(3) Now by assumption in (2), c is not an endpoint since f(a) and f(b) =0 and we are assuming that f(c) is positive. So, we know that c ∈ [a,b]

(4) Since by (2), c is a local maximum (see clarification here for definition), we know by an earlier theorem that f'(c) = 0 (see the Theorem here).

(5) Now, if f has any negative values, then let c = its minimum value over the closed interval [a,b]

(6) Now by assumption (5), c is not an endpoint since f(a) and f(b) = 0 and we are assuming that f(c) is negative. So, we know that c ∈ [a,b]

(7) This will be a local minimum (see clarification here for definition) and we know by an earlier theorem that f'(c) = 0 (see the Corollary here).

(8) If there are no positive or negative values for the function f in [a,b], then we can take any point ∈ [a,b] and know that f(c)=0. We further know that f'(c)=0 since in this case f(x)=C (see Lemma 1 here for the Derivative of f(x)=C).

QED

3 comments :

jagadeesh said...

Beautiful blog here. I have a clarification on the proof, though! First case 'c' could be a local "maxima" right?

Once again, thank you for such a beautiful blog.

Larry Freeman said...

Hi Jagadeesh,

Thanks for noticing that. Yes, that's a typo. First case 'c' should have read local maxima.

I've updated the blog.

Cheers,

-Larry

Deepak Suwalka said...

It's a nice post about Rolle's Theorem. I really like it. It's really helpful. Thanks for sharing it.