WHAT DOES
f
0
SAY ABOUT
f
Since
f
0
(
x
) represents the slope of the curve
y
=
f
(
x
), we can find the direction
in which the curve proceed at each point. Thus, we can find some information about
f
(
x
) from information about
f
0
(
x
). In particular, we will see how the derivative of
f
can tell us where
f
is increasing or decreasing.
(1) If
f
0
(
x
)
>
0 on an interval, then
f
is increasing on that interval.
(2) If
f
0
(
x
)
<
0 on an interval, then
f
is decreasing on that interval.
Example 1.
The graph of a function
f
0
is given. What can we say about
f
?
Solution
From Figure 1, we can see that
f
0
(
x
) is negative when 0
< x <
1. So
the original function
f
must be decreasing on the interval (0
,
1). Similarly,
f
0
(
x
)
is positive for
x <
0 and
x >
1, so
f
is increasing on the intervals (
∞
,
0) and
(0
,
∞
).
Also note that
f
has horizontal tangents when
x
= 0 and
x
= 1, since
f
0
(0) =
f
0
(1) = 0.
We say that
f
in the above example has a
local maximum
at 0 and
local minimum
at 1.
Since
f
00
is the derivative of
f
0
, if
f
00
is positive, then
f
0
is an increasing function.
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 Fall '09
 BenedictGross
 Derivative, Slope, Convex function

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