# 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 $f\left(x\right)={\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x$
differentiating wrt x:
$f\text{'}\left(x\right)=4{\mathrm{sin}}^{3}x\mathrm{cos}x+4{\mathrm{cos}}^{3}x\left(-\mathrm{sin}x\right)\phantom{\rule{0ex}{0ex}}=4\mathrm{sin}x\mathrm{cos}x\left({\mathrm{sin}}^{2}x-{\mathrm{cos}}^{2}x\right)\phantom{\rule{0ex}{0ex}}=-2\left(2\mathrm{sin}x\mathrm{cos}x\right)\mathrm{cos}2x\phantom{\rule{0ex}{0ex}}=-2\mathrm{sin}2x\mathrm{cos}2x\phantom{\rule{0ex}{0ex}}f\text{'}\left(x\right)=-\mathrm{sin}4x$
we have : $x\in \left[0,\frac{\pi }{2}\right]⇒4x\in \left[0,2\mathrm{\pi }\right]$

therefore

so f(x) is decreasing on $\left[0,\frac{\mathrm{\pi }}{4}\right]$
f'(x)>0 for $\frac{\mathrm{\pi }}{4}
so f(x) is increasing on $\left[\frac{\pi }{4},\frac{\pi }{2}\right]$

hope this helps you

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