Section 4.1
:
Key concepts?
..?..
Absolute (or global) vs. relative (or local) maxima and
minima, critical numbers
A function
f
with domain
D
has a
global maximum
at
c
if
f
(
c
)
≥
f
(
x
) for all
x
in the domain of f
.
We call
f
(
c
) the
maximum value
of
f
on
D
.
We say
f
has a
local
maximum
at
c
if
f
(
c
)
≥
f
(
x
) for all
x
in some suitably small
neighborhood of c
(note: this requires that
f
(
x
) is defined in
some neighborhood of
c
, so that in particular
c
cannot be an
endpoint of
D
).
Global and local minima are defined in the same way,
using
≤
instead of
≥
.
Note that the function
f
(
x
) =
x
with domain [0,1] has no
local maximum or minimum!
It does however have a
global maximum (at
c
=1) and a global minimum (at
c
=0).
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0.2
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0.8
1.0
0.2
0.4
0.6
0.8
1.0
The function
f
(
x
) =
x
with domain (0,1) has no global or
local maxima or minima!
Ditto for the function
f
(
x
) =
x
with domain
R
.
The function
f
(
x
) = cos
x
with domain
R
has global maxima
at all points
c
=2
π
n
; it also has local maxima there.
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 Fall '11
 Staff
 Calculus, Extreme value, global maximum, Rolle's theorem

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