find the maximum and the minimum values of the following functions over R 3sinx-4cosx
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Answer:
y
=
3
sin
x
+
4
cos
x
+
2
First express
3
sin
x
+
4
cos
x
as
a
cos
(
x
+
B
)
So, you want to look for the
a
,
A
and
B
⇒
3
sin
x
+
4
cos
x
=
a
cos
(
x
+
A
)
⇒
3
sin
x
+
4
cos
x
=
a
cos
x
cos
A
−
a
sin
x
sin
A
Equate the coefficients of
sin
x
and
cos
x
⇒
a
cos
A
=
4
−
a
sin
A
=
3
⇒
a
sin
A
=
−
3
a
sin
A
a
cos
A
=
−
3
4
⇒
tan
A
=
−
3
4
⇒
A
≅
−
36.87
º
and
(
a
cos
A
)
2
+
(
a
sin
A
)
2
=
(
4
)
2
+
(
−
3
)
2
⇒
a
2
=
25
⇒
a
=
5
Hence,
⇒
3
sin
x
+
4
cos
x
=
5
cos
(
x
−
36.87
º
)
Thus,
y
=
3
sin
x
+
4
cos
x
+
2
=
5
cos
(
x
−
36.87
º
)
+
2
The maximum value of
y
occurs when
cos
x
=
1
y
max
=
5
(
1
)
+
2
=
7
The minimum value of
y
occurs when
cos
x
=
−
1
y
min
=
5
(
−
1
)
+
2
=
−
3
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