find the maximum and minimum value over R f( x)=13cosx+3√3sinx_4
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Answered by
1
Answer:
Let,
f
(
x
)
=
13
cos
x
+
3
√
3
sin
x
−
4
.
∴
f
(
x
)
=
14
(
13
14
cos
x
+
3
√
3
14
sin
x
)
−
4
...
...
...
...
(
⋆
)
.
Note that,
(
13
14
)
2
+
(
3
√
3
14
)
2
=
169
14
2
+
27
14
2
=
196
14
2
=
1
.
This means that, the point
(
13
14
,
3
√
3
14
)
lies on the
unit circle :
x
2
+
y
2
=
1
.
Consequently,
∃
a unique
θ
∈
[
0
,
2
π
)
, such that,
cos
θ
=
13
14
and
sin
θ
=
3
√
3
14
...
...
...
...
(
⋆
⋆
)
.
Utilising this fact in
(
⋆
)
, we have,
f
(
x
)
=
14
{
cos
θ
cos
x
+
sin
θ
sin
x
}
−
4
,
or
,
f
(
x
)
=
14
cos
(
x
−
θ
)
−
4
.
But,
∀
x
∈
R
,
−
1
≤
cos
(
x
−
θ
)
≤
1
.
Multiplying by
14
>
0
,
−
14
≤
14
cos
(
x
−
θ
)
≤
14
,
x
∈
R
.
Adding
−
4
,
−
18
≤
14
cos
(
x
−
θ
)
−
4
≤
10
,
x
∈
R
.
⇒
∀
x
∈
R
,
−
18
≤
f
(
x
)
≤
10
.
⇒
f
min
=
−
18
and
f
max
=
10
.
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