Explanation

(b)
$f(x)\sin^{4}{x + \cos^{4}x}$
$= (\sin^{2}{x + \cos^{2}{x\left. \ \ \right)^{2}} - 2\sin^{2}{x\cos^{2}x}}$
$= 1 - \frac{\sin^{2}{2x}}{2} = 1 - \left( \frac{1 - \cos{4x}}{4} \right)$
$= \frac{3}{4} + \frac{1}{4}\cos{4x}$
For $f(x)to\ be\ increasing$, $f^{'}(x) > 0$
$\Rightarrow - \sin{4x > 0}$
$\Rightarrow \sin{4x < 0}$
$\Rightarrow \pi < 4x < \frac{3\pi}{2}$
$\Rightarrow \frac{\pi}{4} < x < \frac{3\pi}{8}$