Prove this as an increasing function 9 Prove that y = 4 sin θ(2 + cos θ) - Maths - Application of Derivatives

Question

Prove this as an increasing function

9     P r o v e   t h a t   y   =   4
  sin   θ ( 2   +   cos   θ )
  -   θ   i s   a n   i n c r e a sin
g   f u n c t i o n   o f   θ   i n  
0 ,   π 2

Abhishek Jha · Accepted Answer

Dear Student,
Here is the solution of your asked query:
We have,Let f(θ)=y=4 sinθ2+cosθ-θf'(θ)=2+cosθ4 cosθ+4sin2θ2+cosθ2-1f'(θ)=8 cosθ+42+cosθ2-1f'(θ)=8cosθ+4-4-cos2θ-4cosθ2+cosθ2f'(θ)=4cosθ-cos2θ2+cosθ2f'(θ)=cosθ4-cosθ2+cosθ2>0 for all θ∈0,π2Hence, f(θ) is increasing on 0,π2
Regards

Prove this as an increasing function 9 . P r o v e t h a t y = 4 sin θ ( 2 + cos θ ) - θ i s a n i n c r e a sin g f u n c t i o n o f θ i n 0 , π 2 .

Prove this as an increasing function

$9 . P r o v e t h a t y = \frac{4 \sin θ}{(2 + \cos θ)} - θ i s a n i n c r e a \sin g f u n c t i o n o f θ i n [0, \frac{π}{2}] .$