# Rolle's theorem

In mathematics, **Rolle's theorem** first stated by Michel Rolle, and published in 1691 states that if a function *f* is continuous on a closed interval [*a*,*b*] and differentiable on the open interval (*a*,*b*), and *f*(*a*) = *f*(*b*) then there is some number *c* in the open interval (*a*,*b*) such that

It states that, for smooth curves, if the function is equal at two points there must be a stationary point somewhere between them. All the assumptions are necessary. For example, if

*f*(

*x*) = |x|, the

absolute value of

*x*, then we have that f(-1) = f(1), but there is no

*x* between -1 and 1 for which

*f* '(

*x*) = 0.

Rolle's Theorem is used in proving the mean value theorem, which can be seen as a generalisation of it.

**Proof of Rolle's Theorem:** The idea of the proof is to argue that if *f*(*a*) = *f*(*b*) then f must attain either a maximum or a minimum somewhere between *a* and *b*, and *f* ' (*x*) = 0 at either of these points.

Now, by assumption, *f* is continuous on [*a* , *b*], and by the continuity property is bounded and attains both its maximum and its minimum at points of [*a* , *b*]. If these are both attained at endpoints of [*a* , *b*] then *f* is constant on [*a* , *b*] and so f ' (x) = 0 at every point of (a , b).

Suppose then that the maximum is obtained at an interior point *x* of (*a* , *b*) ( the argument for the minimum is very similar). We wish to show that *f* ' (*x*) = 0. We shall examine the left-hand and right-hand derivatives separately.

For *y* just below *x*, ( *f*(*x*) - *f*(*y*) ) / (*x* - *y*) is non-negative, since *x* is a maximum. Thus the limit lim_{y->x-} is non-negative. (Note that we assume that *f* is differentiable to guarantee that the left-hand and right-hand derivatives exist; it does not follow from the other assumptions).

For *y* just greater than *x*, ( *f*(*x*) - *f*(*y*) ) / (*x* - *y*) is non-positive. Thus lim_{y->x+} is non-positive.

Finally, since *f* is differentiable at *x*, these two limits must be equal and hence are both 0. This implies that *f* ' (*x*) = 0.

**Generalization:** The theorem is usually stated in the form above, but it is actually valid in a slightly more general setting:
We only need to assume that *f* : [*a* , *b*] `->` **R** is continuous on [*a* , *b*], that *f*(*a*) = *f*(*b*), and that for every *x* in (*a* , *b*) the limit lim_{h->0} (*f*(*x*+*h*)-*f*(*x*))/*h* exists or is equal to +/- infinity.

## References