Rolle's Theorem: What is it, and how do I use it?

[Thor]

I was absent the day we were tought about this. I need to know what it is and how to use it. We don't use books, and I've looked online, but have only found really hard to understand descriptions. Any help would be GREATLY appreciated, thanks!

 

[pka]

ROLLE’S Theorem: If the function f is continuous on [a,b], differentiable on (a,b) and f(a)= f(b) then there is a point c is (a,b) such that f’(c)=0.

 

[galactus]

Here's the 'mathy' definition of Rolle's theorem:

"Let f be differentiable on (a,b) and continuous on [a,b]. If f(a)=f(b)=0, then there is at least one point c in (a,b) where f'(c)=0".

What that means is between any two points, a and b, where a curve crosses the x-axis, there is at least one place where the tangent line to the curve is horizontal.

Let me illustrate with a graph. See on the graph the points 2 and 4 where f(x)=0?. Where the graph crosses the x-axis. The theorem says there is at least one point between those 2 points, a=2 and b=4, where a tangent line is horizontal. That is, the slope is 0. In this case, that point is c=3.
The x-axis is horizontal. Well, there is at least one point in that interval where a tangent line is parallel to the x-axis.

See?. It's not complicated. It's just that all the 'Mathspeak' can make it more confusing than it is.

 

[Thor]

Great! Wodnerful! Thank you so much! That has to be the easiest definition that I've ever seen! Thanks!