Find the intervals in which the function f given by f(x)=4sinx−2x−xcosx2+cosx is (i) increasing (ii) decreasing.
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Solution
a function f(x) is increasing if f'(x)>0 and decreasing if f'(x)<0 f(x)=4sinx−2x−xcosx2+cosxf′(x)=xsinx+3cosx−22+cosx+(sinx)(4sinx−2x−xcosx)(2+cosx)2=4sin2x+3cos2x+4cosx−4(cosx+2)24(1−cos2x)+3cos2x+4cosx−4(cosx+2)2=4cosx−cos2x(cosx+2)2=cosx(4−cosx)(2+cosx)2 f′(x)=cosx(4−cosx)(2+cosx)2 Sign f′(x) only depends on cosx because (4−cosx) and (2+cosx)2 are always greater than 0. f′(x) when cosx>0 and f′(x)<0 when cosx<0