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You can see that
f
(1) = 5 is a local maximum, whereas the absolute
maximum is
f
(1) = 37.
s
This absolute maximum
is not a local maximum
because it occurs at
an endpoint.
Also,
f
(0) = 0 is a local minimum and
f
(3) = 27 is both a local and
an absolute minimum.
s
Note that
f
has neither a local
nor an absolute maximum at
x
= 4.
Example 4 (4.1)
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View Full Document If
f
(
x
) =
x
3
, then
f ’
(
x
) = 3
x
2
, so
f ’
(0) = 0.
s
However,
f
has no maximum or minimum at 0—as you
can see from the graph.
s
Alternatively, observe that
x
3
> 0 for
x
> 0 but
x
3
< 0 for
x
< 0.
The fact that
f ’
(0) = 0 simply means
that the curve
y = x
3
has a horizontal
tangent at (0, 0).
s
Instead of having a maximum
or minimum at (0, 0), the curve
crosses its horizontal tangent there.
Example 5 (4.1)
FERMAT’S THEOREM
f
(
x
) = 
x
 has its (local and
absolute) minimum value at 0.
s
However, that value can’t be found by setting
f ’
(
x
) = 0.
s
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This note was uploaded on 05/23/2011 for the course MAC 2311 taught by Professor Noohi during the Fall '08 term at FSU.
 Fall '08
 Noohi
 Calculus

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