Find the intervals in which the function f(x) = sin^4x + cos^4x is increasing or decreasing where the range of x is ( 0, pie/2) .

the given function is f(x)=sin4x+cos4x
differentiating wrt x:
f'(x)=4sin3xcosx+4cos3x(-sinx)=4sinxcosx(sin2x-cos2x)=-2(2sinxcosx)cos2x=-2sin2xcos2xf'(x)=-sin4x
we have : x[0,π2]4x[0,2π]
sin4x0 for x[0,π4] andsin4x0 for x[π4,π2]sin4x0 for x[0,π4] andsin4x0 for x[π4,π2]
therefore
f'(x)<0 for 0<x<π4
so f(x) is decreasing on [0,π4]
f'(x)>0 for π4<x<π2
so f(x) is increasing on [π4,π2]

hope this helps you

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