discuss maxima and minima of tan x - 2 x
Answers
Answered by
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Step-by-step explanation:
Explanation:
Let us take
[
0
,
2
π
]
,
as the Domain of the given function.
We know that, for maxima/minima,
d
y
d
x
=
0
.
Also, for maxima,
d
2
y
d
x
2
<
0
,
&
>
0
for minima.
y
=
tan
x
−
2
x
∴
d
y
d
x
=
sec
2
x
−
2
,
&
,
d
2
y
d
x
2
=
2
sec
2
x
tan
x
.
Now,
d
y
d
x
=
0
∴
sec
2
x
=
2
∴
sec
x
=
±
√
2
.
sec
x
=
√
2
∴
x
=
π
4
;
7
π
4
,
&
,
sec
x
=
−
√
2
∴
x
=
3
π
4
;
5
π
4
.
[
d
2
y
d
x
2
]
x
=
π
4
=
2
(
√
2
)
2
(
1
)
=
4
>
0
.
⇒
y
min
(
π
4
)
=
tan
(
π
4
)
−
2
(
π
4
)
=
1
−
π
2
,
and, similarly,
∵
,
[
d
2
y
d
x
2
]
x
=
5
π
4
=
4
>
0
,
∴
,
y
min
(
5
π
4
)
=
1
−
5
π
2
.
[
d
2
y
d
x
2
]
x
=
3
π
4
=
2
(
√
2
)
2
(
−
1
)
=
−
4
<
0
,
and,
[
d
2
y
d
x
2
]
x
=
7
π
4
=
−
4
,
∴
y
max
(
3
π
4
)
=
−
1
−
3
π
2
,
&
,
y
max
(
7
π
4
)
=
−
1
−
7
π
2
.
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