Thursday, October 13
−
Lecture 16 :
Maximum and minimum values of a function.
(Refers to Section 3.4 and 4.4 in your text)
Students who have mastered the content of this lecture know about
:
Local maxima and minima of a
function f, absolute maxima and minima of a function f, extreme values of f, the EVT, differentiable
functions being continuous, critical points, Fermat’s theorem.
Students who have practiced the techniques presented in this lecture will be able to
:
Find the local
maxima and minima of a function by first locating critical points, find the absolute maximum and
minimum of a function by first locating critical points,
16.1
Definition
−
Let
f
be a function with domain
D
. We say that
f
has an
absolute
(global) maximum
at the number
c
in
D
if
f
(
x
)
≤
f
(
c
) for all
x
in
D
. We say that
f
has an
absolute (global) minimum
at
m
in
D
if
f
(
m
)
≤
f
(
x
) for all
x
in
D
.
In this case we say
that
f
(
m
) and
f
(
c
) are
extreme values
of
f
on its domain
D
.
16.2
Definition
Let
f
be a function with domain
D
. Then we say that
f
has a
local
maximum
at the number
c
in
D
if
f
(
x
)
≤
f
(
c
) for all
x
in an open interval centered at
c
which intersects
D
.
We say that
f
has a
local minimum
at the number
m
in
D
if
f
(
x
)
≤
f
(
m
) for all
x
in an open interval centered at
m
which intersects
D
.

Note that a local maximum or minimum may be an “
endpoint
”.

It may also be on a part of the curve where there is not derivative: consider the curve
of
f
(
x
) = 
x
 at
x
= 0 for example.
16.3
Example – Consider the function
with domain
D
= [
−
1, 4] whose graph is plotted below.
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View Full DocumentWith this graph the best we can do is approximate the absolute maximum and minimum.

We estimate that
f
(
x
) has an
absolute maximum
at
x
=
−
1 where the point (
x
,
f
(
x
) ) =
(
−
1, 37) is the absolute maximum

We estimate that
f
(
x
) has an
absolute minimum
at
x
= 3 where the point (
x
,
f
(
x
) ) = (3,
−
27)
is the absolute minimum.

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 Spring '11
 RobertAndre
 Calculus

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