The mean value theorem is the most important theorem about the derivative of a
function of one variable. It pertains only to a real valued function of a real variable. The
best versions of many other theorems depend on this fundamental result. The
mean value theorem is based on the following special case known as Rolle’s
theorem^{1} .
It is an existence theorem and like the other existence theorems in analysis, it depends on
the completeness axiom.
Theorem 5.9.1Suppose f :
[a,b]
→ ℝ is continuous,
f (a) = f (b),
and
f : (a,b) → ℝ
has a derivative at every point of
(a,b)
. Then there exists x ∈
(a,b)
such that f^{′}
(x)
=
0.
Proof: Suppose first that f
(x)
= f
(a)
for all x ∈
[a,b]
. Then any x ∈
(a,b)
is a
point such that f^{′}
(x)
= 0. If f is not constant, either there exists y ∈
(a,b)
such that
f
(y)
> f
(a)
or there exists y ∈
(a,b)
such that f
(y)
< f
(b)
. In the first case, the
maximum of f is achieved at some x ∈
(a,b)
and in the second case, the minimum of f
is achieved at some x ∈