8.1. Find the maximum value of the expression $\frac{\sin \left(x+\frac{\pi}{4}\right)}{2 \sqrt{2}(\sin x+\cos x) \cos 4 x-\cos 8 x-5}$.
Answer: 0.5. Solution. Let's estimate the modulus of the given expression in the condition:

$$
\begin{aligned}
& \left|\frac{\sin \left(x+\frac{\pi}{4}\right)}{2 \sqrt{2}(\sin x+\cos x) \cos 4 x-\cos 8 x-5}\right| \leq \frac{1}{5-2 \sqrt{2}(\sin x+\cos x) \cos 4 x+\cos 8 x} \\
& =\frac{1}{4-4 \sin \left(x+\frac{\pi}{4}\right) \cos 4 x+2 \cos ^{2} 4 x}=\frac{1}{4+2\left(\cos 4 x-\sin \left(x+\frac{\pi}{4}\right)\right)^{2}-2 \sin ^{2}\left(x+\frac{\pi}{4}\right)} \leq \frac{1}{2} \\
& \text { The upper bound can be achieved if }\left\{\begin{array}{l}
\cos 4 x=\sin \left(x+\frac{\pi}{4}\right), \\
\sin \left(x+\frac{\pi}{4}\right)=-1,
\end{array}\right. \text { and indeed }
\end{aligned}
$$

it is achieved at $x=\frac{5 \pi}{4}+2 \pi k$.