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Consider the functionf(x)=(4sinxxcox2x)(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.

 Feb 23, 2016
 #1
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Consider the function f(x)=4sin(x)xcos(x)2x2+cos(x) for 0x2π

 

1) Show that f'(x)=4cosx-cos^2x/(2+cosx)^2

 

 f(x)=4sin(x)xcos(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)=uvf(x)=uvuvv2u=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)]21f(x)=8cos(x)+4cos2(x)+4sin2(x)[2+cos(x)]21f(x)=8cos(x)+4[cos2(x)+sin2(x)][2+cos(x)]21cos2(x)+sin2(x)=1f(x)=8cos(x)+4[2+cos(x)]21f(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)+444cos(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. 

 

laugh

 Feb 23, 2016
edited by heureka  Feb 23, 2016
edited by heureka  Feb 23, 2016

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