Question

Find the intervals in which the function $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x\forall x\in [0,\pi /2]$ is increasing and decreasing.

Answer 1

${\sin }^{4}x+{\cos }^{4}x$

$4{\sin }^{3}x\cos x-4{\cos }^{3}x\sin x>0$

$\Rightarrow \sin x\cos x[{\sin }^{2}x-{\cos }^{2}x]>0$

$\Rightarrow \frac{\sin 2x}{2}[{\sin }^{2}x-{\cos }^{2}x]>0$

$\Rightarrow \frac{2\sin 2x\cos 2x}{2\times 2}<0$

$\Rightarrow \frac{\sin 4x}{4}<0$

$xϵ[0,\frac{x}{2}]>4x=[0,2\pi ]$

For increasing $\Rightarrow x\epsilon [\frac{\pi }{4},\frac{\pi }{2}]$

for decreasing

$xϵ[0,\frac{\pi }{4}]$