Let $f(x)=\sin ^4 x+\cos ^4 x$. Then f is increasing function in the interval 

$\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

Let $f(x)=\sin ^4 x+\cos ^4 x$. Then,

$f'(x) =4 \sin ^3 x\cos x-4 \cos ^3 x \sin x$

$\Rightarrow f'(x) =-4 \cos x \sin x\left(\cos ^2 x-\sin ^2 x\right)$

$\Rightarrow f'(x)=-2 \sin 2 x \cos 2 x=-\sin 4 x$

For f(x) to be increasing, we must have

$f'(x)>0 \Rightarrow-\sin 4 x>0 \Rightarrow \sin 4 x<0$

$\Rightarrow 4 x \in(\pi, 2 \pi) \Rightarrow x \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$