Calculus/Rolle's Theorem
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If a function,
, is continuous on the closed interval ![[a,b] \](http://upload.wikimedia.org/math/c/f/c/cfcbcc2c2a41716ab844e25069e87453.png)
, is differentiable on the open interval 
, and 
, then there exists at least one number c, in the interval such that 
Rolle's Theorem is important in proving the Mean Value Theorem.
[edit] Examples
Example:
f(x) = x2 − 3x. Show that Rolle's Theorem holds true somewhere within this function. To do so, evaluate the x-intercepts and use those points as your interval.
Solution:
1: The question wishes for us to use the x-intercepts as the endpoints of our interval.
Factor the expression to obtain x(x − 3) = 0. x = 0 and x = 3 are our two endpoints. We know that f(0) and f(3) are the same, thus that satisfies the first part of Rolle's theorem (f(a) = f(b)).
2: Now by Rolle's Theorem, we know that somewhere between these points, the slope will be zero. Where? Easy: Take the derivative.
= 2x − 3
Thus, at x = 3 / 2, we have a spot with a slope of zero. We know that 3 / 2 (or 1.5) is between 0 and 3. Thus, Rolle's Theorem is true for this (as it is for all cases). This was merely a demonstration.
