Let f(x)= sin^4⁡x+ cos^4⁡x. Then, f is an increasing function in the interval:

$$f(x)={\mathrm{sin}}^{4}⁡x+{\mathrm{cos}}^{4}⁡x$$f(x)=sin4⁡x+cos4⁡x

$${f}^{'}(x)=4{\mathrm{sin}}^{3}⁡x\mathrm{cos}⁡x+4{\mathrm{cos}}^{3}⁡x(-\mathrm{sin}⁡x)$$f′(x)=4sin3⁡xcos⁡x+4cos3⁡x(−sin⁡x)

$$=4\mathrm{sin}⁡x\mathrm{cos}⁡x({\mathrm{sin}}^{2}⁡x-{\mathrm{cos}}^{2}⁡x)$$=4sin⁡xcos⁡x(sin2⁡x−cos2⁡x)

$$= -2\mathrm{sin}⁡2x\mathrm{cos}⁡2x$$=−2sin⁡2xcos⁡2x

$$= -\mathrm{sin}⁡4x$$=−sin⁡4x

$f(x)$f(x), is increasing when ${f}^{'}(x)>0$f′(x)>0

$\Rightarrow$⇒ $-\mathrm{sin}⁡4x>0$−sin⁡4x>0

$\Rightarrow$⇒ $\mathrm{sin}⁡4x<0$sin⁡4x<0

$$\Rightarrow \pi <4x<2\pi$$⇒π<4x<2π

$$\Rightarrow \frac{\pi }{4}<x<\frac{\pi }{2}$$⇒4π​<x<2π​

$\Rightarrow$⇒ $x \in (\frac{\pi }{4},\frac{\pi }{2})$x∈(4π​,2π​)