The same logic applies to Figure M.102(b). In (b), to the left of
x
= 2 (in Zone III), the
slope is positive, but to the right of
x
= 2 (in Zone IV), it is negative. That is, the slope is
steadily
decreasing
in value as
x
increases: the “slope of the slope,” the second deriva
tive, is
negative
. From Table M.102, we have that the second derivative of the function
in (b) is a constant –2 < 0.
If the second derivative of a function is
negative
at a point
where the first derivative is zero, then at this point, the function has a (local)
maxi
mum
.
One trick that may prove helpful in remembering the sign of the second derivative
is the following. The graph in Figure M.102(a) may remind you of a
grin
(and a grin is
a “positive” expression). The graph in Figure M.102 (b) may remind you of a
frown
(and a frown is a “negative” expression). The trick may seem silly, but it has the advan
tage that it is so silly that it is difficult to forget.
Two final points should be noted, with the help of Figure M.103. First, we have used
the expression “
local
” maximum and minimum. Point
A
in Figure M.103(a) is a local
maximum point, since the value of the function at
A
is greater than for any points in the
M106
MATH MODULE 10: CALCULUS RESULTS FOR THE NONCALCULUS SPEAKER
(b)
y
=
x
3
–
27
x
O
D
C
y
x
(a)
O
A
•
•
•
•
(c)
y
=
x
3
O
y
x
B
y
x
FIGURE M.103
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immediate neighbourhood of
A.
Yet the value of the function at
B
is greater than at
A
(and in fact at
any
other point).
B
is therefore referred to as a
global maximum
. (Of course,
any global maximum is also a local maximum.) Figure M.103(b) illustrates a case
where point
C
is a local maximum and
D
is a local minimum. In Figure M.103(b), how
ever, there is
no
finite global maximum or minimum, because the function goes to
minus infinity on the left and to plus infinity on the right.
The second point is illustrated by Figure M.103(c). It is possible for the first deriva
tive of a function to be equal to zero at a point that is neither a maximum nor a mini
mum. At the origin, the derivative of the function
y
=
x
3
is
dy/dx
= 3
x
2
= 3(0) = 0, but the
origin is clearly not a local maximum or minimum. Indeed, the function
y
=
x
3
has nei
ther a local maximum nor a local minimum at
any
point. Yet the origin is a significant
point. The second derivative of the function (the derivative of the derivative) = 6
x
.
Hence its value is negative for
x
< 0, zero for
x
= 0, and positive for
x
> 0. This means
that the function is becoming increasingly
less steep
as it approaches the origin from the
left, and increasingly
steeper
as it moves away from the origin to the right. A point
where this change in curvature occurs, such as the origin in Figure M.103(c), is called
a
point of inﬂection
. Such inﬂection points (as shown in Figures 94 and 102 in the text)
play an important role in the theory of production and costs.
2. Exercises
1. For the following functions, which have the form
y
=
f
(
x
), give the expressions for
the first and second derivatives, and determine whether (and if so where) they
attain local maxima or minima or inﬂection points. Calculate the values for
y
,
dy/dx
,
and
d
2
y
/
dx
2
at these points.
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 Fall '12
 Danvo
 Calculus, Derivative

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