Mean Value Theorem
MATH 464/506, Real Analysis
J. Robert Buchanan
Department of Mathematics
Summer 2007
J. Robert Buchanan
Mean Value Theorem
Extrema
Definition
A function f : I → R is said to have a relative maximum
[respectively relative minimum] at c ∈ I if there exists a
neighborhood Vδ (c) of c such that f (x) ≤ f (c) [respectively
f (x) ≥ f (c)] for all x ∈ Vδ (c) ∩ I. We say that f has a relative
extremum at c ∈ I if it has either a relative maximum or relative
minimum at c.
J. Robert Buchanan
Mean Value Theorem
Interior Extremum Theorem
Theorem (Interior Extremum Theorem)
Let c be an interior point of the interval I at which f : I → R has
a relative extremum. If the derivative of f at c exists, then
f ′ (c) = 0.
Proof.
Corollary
Let f : I → R be continuous on an interval I and suppose that f
has a relative extremum at an interior point c of I. Then either
the derivative of f at c does not exist, or it is equal to zero.
J. Robert Buchanan
Mean Value Theorem
Interior Extremum Theorem
Theorem (Interior Extremum Theorem)
Let c be an interior point of the interval I at which f : I → R has
a relative extremum. If the derivative of f at c exists, then
f ′ (c) = 0.
Proof.
Corollary
Let f : I → R be continuous on an interval I and suppose that f
has a relative extremum at an interior point c of I. Then either
the derivative of f at c does not exist, or it is equal to zero.
J. Robert Buchanan
Mean Value Theorem
Interior Extremum Theorem
Theorem (Interior Extremum Theorem)
Let c be an interior point of the interval I at which f : I → R has
a relative extremum. If the derivative of f at c exists, then
f ′ (c) = 0.
Proof.
Corollary
Let f : I → R be continuous on an interval I and suppose that f
has a relative extremum at an interior point c of I. Then either
the derivative of f at c does not exist, or it is equal to zero.
J. Robert Buchanan
Mean Value Theorem
Rolle’s Theorem
Theorem (Rolle’s Theorem)
Suppose that f is continuous on a closed interval I = [a, b], that
the derivative f ′ exists at every point of the open interval (a, b),
and that f (a) = f (b) = 0. Then there exists at least one point c
in (a, b) such that f ′ (c) = 0.
Proof.
J. Robert Buchanan
Mean Value Theorem
Rolle’s Theorem
Theorem (Rolle’s Theorem)
Suppose that f is continuous on a closed interval I = [a, b], that
the derivative f ′ exists at every point of the open interval (a, b),
and that f (a) = f (b) = 0. Then there exists at least one point c
in (a, b) such that f ′ (c) = 0.
Proof.
J. Robert Buchanan
Mean Value Theorem
Mean Value Theorem
Theorem (Mean Value Theorem)
Suppose that f is continuous on a closed interval I = [a, b] and
that the derivative f ′ exists at every point of the open interval
(a, b). Then there exists at least one point c in (a, b) such that
f (b) − f (a) = f ′ (c)(b − a).
Proof.
J. Robert Buchanan
Mean Value Theorem
Mean Value Theorem
Theorem (Mean Value Theorem)
Suppose that f is continuous on a closed interval I = [a, b] and
that the derivative f ′ exists at every point of the open interval
(a, b). Then there exists at least one point c in (a, b) such that
f (b) − f (a) = f ′ (c)(b − a).
Proof.
J. Robert Buchanan
Mean Value Theorem
Applications of the Mean Value Theorem
Theorem
Suppose that f is continuous on the closed interval I = [a, b],
that f is differentiable on the open interval (a, b), and that
f ′ (x) = 0 for x ∈ (a, b). Then f is constant on I.
Proof.
Corollary
Suppose that f and g are continuous on the closed interval
I = [a, b], that they are differentiable on the open interval (a, b),
and that f ′ (x) = g ′ (x) for x ∈ (a, b). Then there exists a
constant C such that f = g + C on I.
J. Robert Buchanan
Mean Value Theorem
Applications of the Mean Value Theorem
Theorem
Suppose that f is continuous on the closed interval I = [a, b],
that f is differentiable on the open interval (a, b), and that
f ′ (x) = 0 for x ∈ (a, b). Then f is constant on I.
Proof.
Corollary
Suppose that f and g are continuous on the closed interval
I = [a, b], that they are differentiable on the open interval (a, b),
and that f ′ (x) = g ′ (x) for x ∈ (a, b). Then there exists a
constant C such that f = g + C on I.
J. Robert Buchanan
Mean Value Theorem
Applications of the Mean Value Theorem
Theorem
Suppose that f is continuous on the closed interval I = [a, b],
that f is differentiable on the open interval (a, b), and that
f ′ (x) = 0 for x ∈ (a, b). Then f is constant on I.
Proof.
Corollary
Suppose that f and g are continuous on the closed interval
I = [a, b], that they are differentiable on the open interval (a, b),
and that f ′ (x) = g ′ (x) for x ∈ (a, b). Then there exists a
constant C such that f = g + C on I.
J. Robert Buchanan
Mean Value Theorem
Increasing or Decreasing Functions
Definition
A function f : I → R is said to be increasing on the interval I if
whenever x1 and x2 in I satisfy x1 < x2 , then f (x1 ) ≤ f (x2 ).
Function f is decreasing on I if the function −f is increasing on
I.
Theorem
Let f : I → R be differentiable on the interval I. Then:
1
f is increasing on I if and only if f ′ (x) ≥ 0 for all x ∈ I.
2
f is decreasing on I if and only if f ′ (x) ≤ 0 for all x ∈ I.
Proof.
J. Robert Buchanan
Mean Value Theorem
Increasing or Decreasing Functions
Definition
A function f : I → R is said to be increasing on the interval I if
whenever x1 and x2 in I satisfy x1 < x2 , then f (x1 ) ≤ f (x2 ).
Function f is decreasing on I if the function −f is increasing on
I.
Theorem
Let f : I → R be differentiable on the interval I. Then:
1
f is increasing on I if and only if f ′ (x) ≥ 0 for all x ∈ I.
2
f is decreasing on I if and only if f ′ (x) ≤ 0 for all x ∈ I.
Proof.
J. Robert Buchanan
Mean Value Theorem
Increasing or Decreasing Functions
Definition
A function f : I → R is said to be increasing on the interval I if
whenever x1 and x2 in I satisfy x1 < x2 , then f (x1 ) ≤ f (x2 ).
Function f is decreasing on I if the function −f is increasing on
I.
Theorem
Let f : I → R be differentiable on the interval I. Then:
1
f is increasing on I if and only if f ′ (x) ≥ 0 for all x ∈ I.
2
f is decreasing on I if and only if f ′ (x) ≤ 0 for all x ∈ I.
Proof.
J. Robert Buchanan
Mean Value Theorem
Remarks
A function f is said to be strictly increasing on an interval
I if for any points x1 and x2 in I with x1 < x2 we have
f (x1 ) < f (x2 ).
A function is increasing at a point if there is a
neighborhood of the point on which the function is
increasing.
If f ′ is strictly positive at a point it is not necessarily true
that f is increasing at that point.
J. Robert Buchanan
Mean Value Theorem
Remarks
A function f is said to be strictly increasing on an interval
I if for any points x1 and x2 in I with x1 < x2 we have
f (x1 ) < f (x2 ).
A function is increasing at a point if there is a
neighborhood of the point on which the function is
increasing.
If f ′ is strictly positive at a point it is not necessarily true
that f is increasing at that point.
J. Robert Buchanan
Mean Value Theorem
Remarks
A function f is said to be strictly increasing on an interval
I if for any points x1 and x2 in I with x1 < x2 we have
f (x1 ) < f (x2 ).
A function is increasing at a point if there is a
neighborhood of the point on which the function is
increasing.
If f ′ is strictly positive at a point it is not necessarily true
that f is increasing at that point.
J. Robert Buchanan
Mean Value Theorem
First Derivative Test
Theorem (First Derivative Test)
Let f be continuous on the interval I = [a, b] and let c be an
interior point of I. Assume that f is differentiable on (a, c) and
(c, b). Then:
1
If there is a neighborhood (c − δ, c + δ) ⊆ I such that
f ′ (x) ≥ 0 for c − δ < x < c and f ′ (x) ≤ 0 for c < x < c + δ,
the f has a relative maximum at c.
2
If there is a neighborhood (c − δ, c + δ) ⊆ I such that
f ′ (x) ≤ 0 for c − δ < x < c and f ′ (x) ≥ 0 for c < x < c + δ,
the f has a relative minimum at c.
Proof.
J. Robert Buchanan
Mean Value Theorem
First Derivative Test
Theorem (First Derivative Test)
Let f be continuous on the interval I = [a, b] and let c be an
interior point of I. Assume that f is differentiable on (a, c) and
(c, b). Then:
1
If there is a neighborhood (c − δ, c + δ) ⊆ I such that
f ′ (x) ≥ 0 for c − δ < x < c and f ′ (x) ≤ 0 for c < x < c + δ,
the f has a relative maximum at c.
2
If there is a neighborhood (c − δ, c + δ) ⊆ I such that
f ′ (x) ≤ 0 for c − δ < x < c and f ′ (x) ≥ 0 for c < x < c + δ,
the f has a relative minimum at c.
Proof.
J. Robert Buchanan
Mean Value Theorem
Inequalities
Example
ex ≥ 1 + x for all x ∈ R.
−x ≤ sin x ≤ x for all x ≥ 0.
(1 + x)α ≥ 1 + αx for all x > −1.
If 0 < α < 1 and x ≥ 0 then x α ≤ αx + (1 − α).
J. Robert Buchanan
Mean Value Theorem
Intermediate Value Property of Derivatives
Lemma
Let I ⊆ R be an interval, let f : I → R, let c ∈ I, and assume that
f has a derivative at c. Then:
1
If f ′ (c) > 0, then there is a number δ > 0 such that
f (x) > f (c) for x ∈ I such that c < x < c + δ.
2
If f ′ (c) < 0, then there is a number δ > 0 such that
f (x) > f (c) for x ∈ I such that c − δ < x < c.
Proof.
J. Robert Buchanan
Mean Value Theorem
Intermediate Value Property of Derivatives
Lemma
Let I ⊆ R be an interval, let f : I → R, let c ∈ I, and assume that
f has a derivative at c. Then:
1
If f ′ (c) > 0, then there is a number δ > 0 such that
f (x) > f (c) for x ∈ I such that c < x < c + δ.
2
If f ′ (c) < 0, then there is a number δ > 0 such that
f (x) > f (c) for x ∈ I such that c − δ < x < c.
Proof.
J. Robert Buchanan
Mean Value Theorem
Darboux’s Theorem
Theorem (Darboux’s Theorem)
If f is differentiable on I = [a, b] and if k is a number between
f ′ (a) and f ′ (b), then there is at least one point c in (a, b) such
that f ′ (c) = k.
Proof.
J. Robert Buchanan
Mean Value Theorem
Darboux’s Theorem
Theorem (Darboux’s Theorem)
If f is differentiable on I = [a, b] and if k is a number between
f ′ (a) and f ′ (b), then there is at least one point c in (a, b) such
that f ′ (c) = k.
Proof.
J. Robert Buchanan
Mean Value Theorem
Example
Example
Define the function

for 0 < x ≤ 1,
 1
g(x) =
0
for x = 0,

−1 for −1 ≤ x < 0.
Function g(x) fails to satisfy the intermediate value property on
[−1, 1], thus by Darboux’s Theorem, g is not the derivative of
any function defined on [−1, 1].
J. Robert Buchanan
Mean Value Theorem
Homework
Read Section 6.2,
Pages 175-176: 1, 4, 5 , 7 , 9, 10, 13, 17
Boxed problems should be written up separately and submitted
for grading at class time on Friday.
J. Robert Buchanan
Mean Value Theorem