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Increasing and decreasing
f(x) = x
2
– 2x + 3
f
′
(x) = 2x – 2
When f’(x)>0 (x>1)
the function is
increasing
When f’(x)<0 (x<1)
the function is
decreasing
0
2
4
6
8
10
12
-2
-1
0
1
2
3
4
f(x) = x
3
– 3x
2
– 9x + 6
2
2
increasin
'(
)
3
6
9
3(
2
3)
3(
3)(
1)
1
(
)(
)
1
3
(
)(
)
3
g
increasi
decreasi
(
)(
)
g
ng
n
f
x
x
x
x
x
x
x
x
x
x
=
−
−
=
−
−
=
−
+
< −
−
−
= +
−
<
<
−
+
= −
<
+
+
= +
f(x) = x
3
– 3x
2
– 9x + 6
-25
-20
-15
-10
-5
0
5
10
15
-3
-2
-1
0
1
2
3
4
5
What if you can’t guess the roots?
2
2
2
4
0 has roots
2
3
6
9
has roots
6
36
4 3 (
9)
6
144
1
2
6
6
b
b
ac
a
x
x
c
x
bx
a
−
−
−
±
−
±
−
⋅
⋅ −
±
=
=
=
+
±
+
Drug concentration, problem 41
2
The concentration of a drug in the
blood strem t hours later
4
( )
3
27
When is th
increasi
is
,
decr
n
?
ea ing
g
s
t
K t
t
=
+
K(t) = 4t/(3t
2
+ 27)
2
2
2
2
2
4(3
27)
4 (6 )
'( )
(3
27)
numerator =
12
108
0
108
wh
increasing
en
t
9
12
0
d
ecreasin
3
g
3
t
t
t
K
t
t
t
t
t
+
−
=
+
−
+
<
>
=
≤
<
>

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