
Mean Value Theorems 9-11
Hence by mean value theorem, there exists c Î (a, b) such that
f ' c
f b f a
b a b a
( )
( ) ( )
.=
−
−
=
−
−
=
0 0
0
Therefore, f ′(x) = 0 has a root x = c.
Note 9.2 In this example, f can have more than one root.
9.3 CAUCHY’S MEAN VALUE THEOREM
Cauchy’s mean value theorem, which is also known as the second mean value theorem or extended
mean value theorem, is the more general form of the mean value theorem and it deals with two
functions f and F.
STATEMENT If two functions f and F are:
a. Both continuous in [a, b]
b. Both differentiable in (a, b)
c. F′(x) ¹ 0 for any x Î (a, b), then there exists at least one real ...