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UPSC NDA Math Model Paper 9
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© examsnet.com
Question : 49
Total: 120
What is the value of p for which the function
f
(
x
)
=
p
sin
x
+
sin
3
x
3
has an extremum at
x
=
π
3
?
0
1
-1
2
Validate
Solution:
Given, the function is
f
(
x
)
=
p
sinx
+
sin
3
x
3
⇒
f
′
(
x
)
=
p
cos
x
+
3
cos
3
x
3
⇒
f
′
(
x
)
=
p
cos
x
+
cos
3
x
⇒
f
′
(
π
3
)
=
p
cos
(
π
3
)
+
cos
3
(
π
3
)
⇒
f
′
(
π
3
)
=
p
cos
(
π
3
)
+
cos
π
The function
f
(
x
)
=
p
sin
x
+
sin
3
x
3
has an extremum at
x
=
π
3
Therefore,
f
′
(
π
3
)
=
0
⇒
p
cos
(
π
3
)
+
cos
π
=
0
⇒
p
2
−
1
=
0
⇒
p
2
=
1
⇒
p
=
2
Hence, the value of
p
for which the function
f
(
x
)
=
p
sin
x
+
sin
3
x
3
has an extremum at
x
=
π
3
is 2.
© examsnet.com
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