4. Let f is continuous on [a, b] and
differentiable on (a, b), and suppose
that f 0(x) 6= 0 on (a, b). If f (a)
and f (b) have opposite sign, then
....
24
THEOREM 26.3:
(The Mean
Value Theorem) Let
f
be con-
tinuous on the closed interval [a, b]
and differentiable on the open interval (a, b). Then there exists at least
one number c ∈ (a, b) such that
f 0(c)
f (b) − f (a)
=
.
b−a
25
Note: The conclusion of the MVT
can be written equivalently as: there
exists at least one number c ∈ (a, b)
such that
f (b) − f (a) = f 0(c)(b − a).
26
Examples:
1.Show that
√
f (x) = 3 x − 4x, 0 ≤ x ≤ 4
satisfies the hypotheses of the MVT
and find the value(s) of c guaranteed by the theorem.
27
2.Does there exist a differentiable
function f that satisfies
f (0) = 2, f (2) = 5
and
f 0(x) ≤ 1 on (0, 2)?
28
Corollary 1: Let f be continuous
on [a, b] and differentiable on (a, b).
If f 0(x) = 0 for all x ∈ (a, b), i.e.,
f 0(x) ≡ 0,
then
f (x) = k,
con-
stant, on [a, b].
29
NOTE: Let
f
be continuous on
[a, b] and differentiable on (a, b). Then
f (x) ≡ k (constant)
if and only if
f 0(x) ≡ 0.
30
Corollary 2: Let f and g be continuous on [a, b] and differentiable
on (a, b). If
f 0(x) = g 0(x)
for all x ∈ (a, b),
then there exists a constant C such
that
f (x) = g(x) + C on [a, b].
31
Def.
A function
f
is strictly
increasing on an interval I if
f (x1) < f (x2) whenever x1 < x2.
f
is strictly decreasing on an in-
terval I if
f (x1) > f (x2) whenever x1 < x2.
32
THEOREM 26.4:
Let
f
be
differentiable on an interval I.
(a) If f 0(x) > 0 for all x ∈ I, then
f is strictly increasing on I.
(b) If f 0(x) < 0 for all x ∈ I, then
f is strictly decreasing on I.
33
Examples:
1. Let
f
be differentiable on an
interval I. If f is strictly increasing
on I, does it follow that f 0(x) > 0
on I?
Consider
f (x) = x3,
g(x) = x + sin x
34
The graph of f (x) = x + sin x
35
THEOREM 26.4’:
Let f
be
continuous on I = [a, b] and differentiable on (a, b).
(a) If f 0(x) > 0 on (a, b), then f
is strictly increasing on [a, b].
(b) If f 0(x) < 0 on (a, b), then f
is strictly decreasing on [a, b].
36
Corollary: Let
f
be continuous
on an interval I
(a) If f 0(x) > 0 on I, except at
isolated points c at which f 0(c) =
0, then f is strictly increasing on
I.
(b) If f 0(x) < 0 on I, except at
isolated points c at which f 0(c) =
0, then f is strictly decreasing on
I.
37
Examples:
1. Find the intervals on which
f (x) =
x3
3 2
− x − 6x + 2
2
is strictly increasing/strictly decreasing.
38
2. Prove that ex > 1 + x for x > 0.
39
3. Prove that x < tan x for 0 <
x < π/2.
40
Def.
A function f is increasing
on an interval I if
f (x1) ≤ f (x2) whenever x1 < x2.
f is decreasing on an interval I if
f (x1) ≥ f (x2) whenever x1 < x2.
41
THEOREM 26.5:
If f is dif-
ferentiable on an interval I, then
(a)
f
is increasing if and only if
f 0(x) ≥ 0 on I.
(b)
f
is decreasing if and only if
f 0(x) ≤ 0 on I.
42
Example:






2 − x2,
2,




x2 − 6x + 11,
Set f (x) = 
x≤0
0<x≤3
x>3
Show that f is differentiable on R.
Find the intervals on which
f
increasing, decreasing.
43
is
f 0 bounded
Let I
be a bounded interval with
endpoints a and b,
and let f :
I → R, be continuous. If:
|f 0(x)| ≤ M for all x ∈ I.
Then:
44
1.
f (a)−M (b−a) ≤ f (x) ≤ f (a)+M (b−a)
for all x ∈ I.
45
2. Let > 0. Then there is a δ > 0
such that
|f (x2) − f (x1)| < whenever |x2 − x1| < δ, x1, x2 ∈ I
46
Def.
f : I → R is uniformly con-
tinuous on
I
if to each
> 0
there is a δ > 0 such that
|f (x2) − f (x1)| < whenever |x2 − x1| < δ, x1, x2 ∈ I.
47
THEOREM 26.6:
Let f : D →
R be continuous. If D is compact,
then f is uniformly continuous on
D.
48
Uniform Continuity
1. Let f : D → R. If D is compact
and
f
is continuous, then
f
is
uniformly continuous.
2. Let
f : I → R, I
an interval.
If f 0 is bounded on I, then f is
uniformly continuous on I.
49
THEOREM 26.7:
Suppose that
f is differentiable on an interval I
and f 0(x) 6= 0 for all x ∈ I. Then f
is one-to-one, f −1 is differentiable
on f (I), and for any b ∈ f (I),
(f −1)0(b)
1
= 0
f (a)
where
b = f (a).
50
Examples:
1. Set
√
f (x) = x + 2 x
on
I =
(0, ∞).
1. Show that f is strictly increasing on I.
2. f has an inverse function. Find
(f −1)0(8).
51
2. Set
f (x) = x − π + cos x
on
I = (0, 2π).
1. Show that f is strictly increasing on I.
2. f has an inverse function. Find
(f −1)0(−1).
52