Consider the functionf(x)=(4sinx−xcox−2x)(2+cosx)for0<=x<=2pi
1)Show that f'(x)=4cosx-cos^2x/(2+cosx)^2
2) identify the interval on which f(x) is increaing or decreaasing.
Consider the function f(x)=4sin(x)−x⋅cos(x)−2x2+cos(x) for 0≤x≤2π
1) Show that f'(x)=4cosx-cos^2x/(2+cosx)^2
f(x)=4sin(x)−x⋅cos(x)−2x2+cos(x)=4sin(x)−x⋅[2+cos(x)]2+cos(x)=4sin(x)2+cos(x)−x⋅[2+cos(x)]2+cos(x)=4sin(x)2+cos(x)−xf(x)=uv⇒f′(x)=u′v−uv′v2u=4sin(x)u′=4cos(x)v=2+cos(x)v′=−sin(x)f′(x)=4cos(x)(2+cos(x))−4sin(x)(−sin(x))[2+cos(x)]2−1f′(x)=8cos(x)+4cos2(x)+4sin2(x)[2+cos(x)]2−1f′(x)=8cos(x)+4[cos2(x)+sin2(x)][2+cos(x)]2−1cos2(x)+sin2(x)=1f′(x)=8cos(x)+4[2+cos(x)]2−1f′(x)=8cos(x)+4−[2+cos(x)]2[2+cos(x)]2f′(x)=8cos(x)+4−[4+4cos(x)+cos2(x)][2+cos(x)]2f′(x)=8cos(x)+4−4−4cos(x)−cos2(x)][2+cos(x)]2f′(x)=8cos(x)−4cos(x)−cos2(x)[2+cos(x)]2f′(x)=4cos(x)−cos2(x)[2+cos(x)]2
2.