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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