There are several important consequences of Rolle's Theorem and
the Mean Value Theorem (MVT).
Mean Value Theorem (MVT). If f is continuous on (a,b) and differentiable on [a,b], then there is an x in (a,b) such that
f′(x) =
f(b) − f(a)
b − a
.
.
Corollary 1, p. 191. If f is defined on an interval I, and f′(x) = 0 on I, then f is constant on I.
This corollary is important in integration - Two functions F,G which have the same derivative on an interval must differ by a
constant C on the interval.
Proof. For any a,b, in I,
f(b) − f(a)
= f′(c)·(b − a) = 0.
Theorem. If f is continuous and differentiable on an interval I, and f′ > 0 on I, then f is
increasing on I.
Proof. For a, b in I,
f(b) − f(a)
= f′(c)·(b −a),
so that f(b) − f(a) has the same sign as b − a.
Note that the conclusion holds if
we assume that f is continuous on I and that for
x not an endpoint of I, f′(x) > 0.
The following is known as the First Derivative Test for a
Local Maximum/Minimum .
Theorem. If a number a in an interval I is a critical point of a function f, and there is a δ > 0, such that for 0 < x − a < δ, f′(x) > 0, and for −δ < x − a < 0, f′(x) < 0, then a is a local minimum point for f on I.
Proof. For 0 < |x − a| < δ,
f(x) − f(a)
= f′(cx)·(x − a)
> 0,
since cx is between a and x.
The Mean Value Theorem may also be used to prove that
derivatives exist.
Theorem. Suppose that f is continuous at a and that limx → a f′(x) exists. Then f′(a) exists and
f′(a) =
lim
x → a
f′(x).
Proof.
lim
x→ a
f(x)−f(a)
x−a
=
lim
x → a
f′(cx)
=
lim
x → a
f′(x).
N.B. - Be Careful! The Theorem does not say that
all functions which are derivatives are also continuous.
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