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Question

Show that the function f(x) = tan x − 4x is decreasing function on (−π/3, π/3).

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Solution

fx=tan x-4xf'x=sec2x-4 =1cos2x-4 =1-4cos2xcos2x =sec2x 1-4cos2x =4sec2x 14-cos2x =4sec2x12-cosx12+cos xHere,-π3<x<π3cos x>12-cosx<-1212-cosx<0 ...1Also,-π3<x<π3cos x>1212+cosx>112+cosx>0 ...24 sec2 x >0 a2>0Now,f'x=4sec2x12-cos x12+cos x<0, -π3, π3 From eqs. (1) and (2)So, fx is decreasing on -π3, π3.

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