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