**Rolle's Theorem**

If a function, , is continuous on the closed interval , 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.

## ExamplesEdit

**Example:**

. 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* = 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.

Thus, at , we have a spot with a slope of zero. We know that (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.