Is #f(x)=sin(3x)-cos(2x-pi)# increasing or decreasing at #x=pi/3#?

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Answer 1 · 2016-07-26T13:50:42

#f darr# at #x=pi/3#.

#f(x)=sin3x-cos(2x-pi)#

But, #cos(-theta)=costheta#

#:. f(x)=sin3x-cos(pi-2x)#

#=sin3x+cos2x,#...........[ as #cos(pi-theta)=-costheta#]

#:. f'(x)=3cos3x-2sin2x#

#:. f'(pi/3)=3cospi-2sin(2pi/3)=3(-1)-2sin(pi-pi/3)#

#=-3-2sin(pi/3)=-3-2(sqrt3/2)=-3-sqrt3<0#

Hence, #f darr# at #x=pi/3#.