Let f x = sin 4 ⁡ x + cos 4 ⁡ x . Then, f is an increasing function in the interval:

Let fx=sin4⁡x+cos4⁡x. Then, f is an increasing function in the interval:
  1. ] 5 π 8 , 3 π 4 [
  2. ] π 2 , 5 π 8 [
  3. ] π 4 , π 2 [
  4. ] 0 , π 4 [

Solution

fx=sin4⁡x+cos4⁡x

f′x=4sin3⁡xcos⁡x+4cos3⁡x-sin⁡x

=4sin⁡xcos⁡xsin2⁡x-cos2⁡x

= -2sin⁡2xcos⁡2x

= -sin⁡4x

fx, is increasing when f′x>0

⇒ -sin⁡4x>0

⇒ sin⁡4x<0

⇒π<4x<2π

⇒π4<x<π2
⇒ x ∈π4,π2

Asked in: JEE Main 2016 (10 Apr Online)

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