Prove that tanx-4x is strictly decreasing in (-pi/3 , pi/3) - Maths - Application of Derivatives

Question

Prove that tanx-4x is strictly decreasing in (-pi/3 , pi/3)

Bhavana Majji · Accepted Answer

Dear student fx=tanx-4xf'x=sec2x-4=1cos2x-4=1-4cos2xcos2x=sec2x1-4cos2x=4sec2x14-cos2x=4sec2x12-cosx12+cosxHere-π312⇒-cosx<-12⇒12-cosx<0 (1)Also-π312⇒12+cosx>1⇒12+cosx>0 24sec2x>0 ∵a2>0Now, f'x=4sec2x12-cosx12+cosx<0 , for all ∈ -π3,π3 from eqs (1) and (2)So f(x) is decreasing on -π3,π3 I hope this helps!! Regards

Prove that tanx-4x is strictly decreasing in (-pi/3 , pi/3).