2.1. Find all values of $x$ for which the greatest of the numbers $x^{2}$ and $\cos 2 x$ is less than $\frac{1}{2}$. In your answer, record the total length of the intervals found on the number line, rounding it to the nearest hundredths if necessary.
Answer: 0.37 (exact value: $\sqrt{2}-\frac{\pi}{3}$).

Solution. The inequality $\max \left\{x^{2}, \cos 2 x\right\}<\frac{1}{2}$ is equivalent to the system of inequalities

$$
\left\{\begin{array}{l}
x^{2}<\frac{1}{2}, \\
\cos 2 x<\frac{1}{2}
\end{array} \Leftrightarrow x \in\left(-\frac{\sqrt{2}}{2} ;-\frac{\pi}{6}\right) \cup\left(\frac{\pi}{6} ; \frac{\sqrt{2}}{2}\right)\right.
$$

The total length of the found intervals is $2\left(\frac{\sqrt{2}}{2}-\frac{\pi}{6}\right)=\sqrt{2}-\frac{\pi}{3} \approx 0.37$.