The Mean Value Theorem
Connect any two points, (a, f (a)) and (b, f (b)), on a differentiable
function f to form a line:
Figure 4.1: 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 4.2: 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
Rolle's Theorem
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 4.3: Rolle's theorem as a case of the Mean Value Theorem
