Math 119 Recitation 7
November 1, 2006
1. Verify that the function
f
(
x
) = sin(2
πx
) satisﬁes
(a)
f
is continuous on [

1
,
1],
(b)
f
is diﬀerentiable on (

1
,
1),
(c)
f
(

1) =
f
(1).
and ﬁnd all numbers
c
∈
(

1
,
1) such that
f
0
(
c
) = 0.
2. Verify that the function
f
(
x
) =
3
√
x
) satisﬁes
(a)
f
is continuous on [0
,
1],
(b)
f
is diﬀerentiable on (0
,
1),
and ﬁnd all numbers
c
∈
(

1
,
1) such that
f
0
(
c
) =
f
(1)

f
(0)
1

0
.
3. Show that the equation 2
x

1

sin
x
= 0 has exactly one real root.
4. Given
f
(
x
) =
x
2
x
+3
,
(a) ﬁnd intervals on which
f
is increasing or decreasing.
(b) ﬁnd the local maximum and minimum values of
f
.
(c) ﬁnd the intervals of concavity and the inﬂection points.
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 Spring '10
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 Math, Calculus, Critical Point, Mathematical analysis, Continuous function

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