Prove that y=( 4 sin theta/ 2+cos theta )- theta is an increasing function of theta in[0, pie/2] NCERT - Maths - Application of Derivatives

Question

Prove that y=( 4 sin theta/ 2+cos theta )- theta is an
increasing function of theta in[0, pie/2] NCERT Q 9,6 2, in this
ques dy/dx is calculated 2 times explain how it was calculated
second time and why?

Vipin Verma · Accepted Answer

y=4sinθ2+cosθ-θHence for the function to be
increasing, y'>0Hence on differentiation we gety'
=4cosθ(2+cosθ)-4sinθ(-sinθ)(2+cosθ)2-1y'
=8cosθ+ 4cos2θ+4sin2θ(2+cosθ)2-1Or y'
=8cosθ+ 4(2+cosθ)2-1Or y' = 8cosθ+ 4-(4
+cos2θ+4cosθ)(2+cosθ)2 y'
=4cosθ-cos2θ(2+cosθ)2So y'
=cosθ(4-cosθ)(2+cosθ)2So cosθ is positive in
(0,π/2), (4-cosθ) >0, as cosθ max value can be 1
only, and the denominator is always positive Hence y' is always
positive in (0,π/2) So y=4sinθ2+cosθ-θ is
increasing in (0,π/2)

Prove that y=( 4 sin theta/ 2+cos theta )- theta is an increasing function of theta in[0, pie/2]. NCERT Q 9,6.2, in this ques dy/dx is calculated 2 times .explain how it was calculated second time and why?