x
y
y
=
f
(
x
)
x
0
ELEMENTARY MATHEMATICS
W W L CHEN and X T DUONG
c
°
W W L Chen, X T Duong and Macquarie University, 1999.
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Chapter 13
APPLICATIONS OF DIFFERENTIATION
13.1. Second Derivatives
Recall that for a function
y
=
f
(
x
), the derivative
f
0
(
x
) represents the slope of the tangent. It is easy to
see from a picture that if the derivative
f
0
(
x
)
>
0, then the function
f
(
x
) is increasing; in other words,
f
(
x
) increases in value as
x
increases. On the other hand, if the derivative
f
0
(
x
)
<
0, then the function
f
(
x
) is decreasing; in other words,
f
(
x
) decreases in value as
x
increases. We are interested in the case
when the derivative
f
0
(
x
) = 0. Values
x
=
x
0
such that
f
0
(
x
0
) = 0 are called stationary points.
Let us introduce the second derivative
f
0
(
x
) of the function
f
(
x
). This is deFned to be the derivative
of the derivative
f
0
(
x
). With the same reasoning as before but applied to the function
f
0
(
x
) instead
of the function
f
(
x
), we conclude that if the second derivative
f
0
(
x
)
>
0, then the derivative
f
0
(
x
)is
increasing. Similarly, if the second derivative
f
0
(
x
)
<
0, then the derivative
f
0
(
x
) is decreasing.
Suppose that
f
0
(
x
0
)=0and
f
0
(
x
0
)
<
0. The condition
f
0
(
x
0
)
<
0 tells us that the derivative
f
0
(
x
) is decreasing near the point
x
=
x
0
. Since
f
0
(
x
0
) = 0, this suggests that
f
0
(
x
)
>
0 when
x
is a
little smaller than
x
0
, and that
f
0
(
x
)
<
0 when
x
is a little greater than
x
0
, as indicated in the picture
below.
†
This chapter was written at Macquarie University in 1999.
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y
y
=
f
(
x
)
x
0
x
y
x
1
x
2
13–2
W W L Chen and X T Duong : Elementary Mathematics
In this case, we say that the function has a local maximum at the point
x
=
x
0
. This means that if we
restrict our attention to real values
x
near enough to the point
x
=
x
0
, then
f
(
x
)
≤
f
(
x
0
) for all such
real values
x
.
LOCAL MAXIMUM.
Suppose that
f
0
(
x
0
)=0
and
f
0
(
x
0
)
<
0
. Then the function
f
(
x
)
has a local
maximum at the point
x
=
x
0
.
Suppose next that
f
0
(
x
0
) = 0 and
f
0
(
x
0
)
>
0. The condition
f
0
(
x
0
)
>
0 tells us that the derivative
f
0
(
x
) is increasing near the point
x
=
x
0
. Since
f
0
(
x
0
) = 0, this suggests that
f
0
(
x
)
<
0 when
x
is a
little smaller than
x
0
, and that
f
0
(
x
)
>
0 when
x
is a little greater than
x
0
, as indicated in the picture
below.
In this case, we say that the function has a local minimum at the point
x
=
x
0
. This means that if we
restrict our attention to real values
x
near enough to the point
x
=
x
0
, then
f
(
x
)
≥
f
(
x
0
) for all such
real values
x
.
LOCAL MINIMUM.
Suppose that
f
0
(
x
0
and
f
0
(
x
0
)
>
0
. Then the function
f
(
x
)
has a local
minimum at the point
x
=
x
0
.
Remark.
These stationary points are called local maxima or local minima because such points may
not maximize or minimize the functions in question. Consider the picture below, with a local maximum
at
x
=
x
1
and a local minumum at
x
=
x
2
.
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