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