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Question

Find the intervals in which the function f given by
f(x)=4sinx2xxcosx2+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)=4sinx2xxcosx2+cosxf(x)=xsinx+3cosx22+cosx+(sinx)(4sinx2xxcosx)(2+cosx)2=4sin2x+3cos2x+4cosx4(cosx+2)24(1cos2x)+3cos2x+4cosx4(cosx+2)2=4cosxcos2x(cosx+2)2=cosx(4cosx)(2+cosx)2
f(x)=cosx(4cosx)(2+cosx)2
Sign f(x) only depends on cosx because (4cosx) and (2+cosx)2 are always greater than 0.
f(x) when cosx>0 and f(x)<0 when cosx<0


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