Connect any two points, (*a*, *f* (*a*)) and (*b*, *f* (*b*)), on a differentiable
function *f* to form a line:

Figure %: Connecting two points on a continuous function

Intuitively, it should be clear that we can find a point *c* between *a* and *b* where the
tangent line is parallel to the secant drawn between *a* and *b*. In other words, it
should be possible to find a point such that the slope of the tangent at *c* is the same as
the slope of the secant line drawn from *a* to *b*.

Figure %: Demonstration of the Mean Value Theorem

This intuitive idea is stated as the Mean Value Theorem, which states that if
*f* is continuous on [*a*, *b*] and differentiable on (*a*, *b*), then there exists a point *c*
on [*a*, *b*] for which

f'(c) = |

Rolle's theorem is a special case of the mean value theorem in which *f* (*a*) = *f* (*b*). It
says: if *f* is continuous on [a,b] and differentiable on (a,b), and *f* (*a*) = *f* (*b*), then there is a
*c* on (*a*, *b*) where *f'*(*c*) = 0.

The figure below should make clear that this is just a special case of the mean value theorem:

Figure %: Rolle's theorem as a case of the Mean Value Theorem

Take a Study Break!