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Mathematics
408
K: Homework
9
Gagan Tara Nanda
18
th
October,
2004
HW
9
:
#
s
6
,
12
,
22
6. Make sure you understand the English carefully here: The idea is to maximize the “rate of growth of
sales”, which in Math means maximize
N
0
(
t
)
.Sowew
ishtomax
im
izethefunct
ion
N
0
(
t
) = 108
t
−
3
t
2
on the interval
1
≤
t
≤
44
.Soa
sa
lway
s
,
f
nd its derivative. We get
N
00
(
t
)=108
−
6
t
,
which we set to zero to get
t
=18
. The derivative exists everywhere, so there are no other critical points.
We need to then compare the values of
N
0
(
t
)
at the endpoints and the critical point. First, let’s check
that we have a local maximum at
t
=18
.U
s
ethe
2
nd
Derivative Test. So
N
000
(
t
)=
−
6
,
which is
>
0
for all
t
.So
N
0
(18)
is a local maximum. We then have
N
0
(1) = 105
,
N
0
(44) =
−
1056
,
N
0
(18) = 972
,
so the maximum rate of growth of sales occurs on day
18
.
12. Let
f
(
x
)=
x
5
+2
x
+5
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