Question
If 2f(sin x)+f(cos x)=x ∀ x ϵ R then range of f(x) is
Answer: Option B
:
B
Put x=sin−1x
2f(x)+f(√1−x2)=sin−1x→(1)
On Putting x=cos−1x
⇒2f(√1−x2)+f(x)=cos−1x→(2)
Eq.(1)×2⇒4f(x)+2f(√1−x2)=2sin−1x→(3)
On subtracting Eq. 2 from Eq. 3 we get -
3f(x)=2sin−1x−cos−1x
f(x)=23sin−1x−13(π2−sin−1x)
=sin−1x−π6
fmax=π2−π6=π3,fmin=−π2−π6=−4π6=−2π3
=[−2π3,π3]
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:
B
Put x=sin−1x
2f(x)+f(√1−x2)=sin−1x→(1)
On Putting x=cos−1x
⇒2f(√1−x2)+f(x)=cos−1x→(2)
Eq.(1)×2⇒4f(x)+2f(√1−x2)=2sin−1x→(3)
On subtracting Eq. 2 from Eq. 3 we get -
3f(x)=2sin−1x−cos−1x
f(x)=23sin−1x−13(π2−sin−1x)
=sin−1x−π6
fmax=π2−π6=π3,fmin=−π2−π6=−4π6=−2π3
=[−2π3,π3]
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