Task 2. (10 points) Find the greatest value of the parameter $m$ for which the inequality $m \sqrt{m}\left(x^{2}-6 x+9\right)+\frac{\sqrt{m}}{\left(x^{2}-6 x+9\right)} \leq \sqrt[4]{m^{3}} \cdot\left|\cos \frac{\pi x}{5}\right|$ has at least one solution.

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# Solution.

1st method.

$m \sqrt{m}\left(x^{2}-6 x+9\right)+\frac{\sqrt{m}}{\left(x^{2}-6 x+9\right)} \leq \sqrt[4]{m^{3}} \cdot\left|\cos \frac{\pi x}{5}\right|$,

$m \sqrt{m}(x-3)^{2}+\frac{\sqrt{m}}{(x-3)^{2}} \leq \sqrt[4]{m^{3}} \cdot\left|\cos \frac{\pi x}{5}\right|$,

$\sqrt[4]{m^{3}}(x-3)^{2}+\frac{1}{\sqrt[4]{m}(x-3)^{2}} \leq\left|\cos \frac{\pi x}{5}\right|$

Applying the Cauchy inequality about the arithmetic mean and geometric mean, we transition to the inequality:

$$
2 \sqrt{\sqrt[4]{m^{3}}(x-3)^{2} \cdot \frac{1}{\sqrt[4]{m}(x-3)^{2}}} \leq \sqrt[4]{m^{3}}(x-3)^{2}+\frac{1}{\sqrt[4]{m}(x-3)^{2}} \leq\left|\cos \frac{\pi x}{5}\right| \leq 1
$$

Solving this inequality, we get

$$
2 \sqrt[4]{m} \leq 1, \sqrt[4]{m} \leq \frac{1}{2}, m \leq \frac{1}{16}
$$

If $m=\frac{1}{16}$, then the inequality $\frac{1}{8}(x-3)^{2}+\frac{2}{(x-3)^{2}} \leq\left|\cos \frac{\pi x}{5}\right|$ has a solution at $x=5$ : $\frac{1}{8} \cdot 4+\frac{2}{4} \leq\left|\cos \frac{5 \pi}{5}\right|, 1 \leq|\cos \pi|$.

## 2nd method.

This is the only value for which the inequality has a solution.

$$
\begin{aligned}
& m \sqrt{m}\left(x^{2}-6 x+9\right)+\frac{\sqrt{m}}{\left(x^{2}-6 x+9\right)} \leq \sqrt[4]{m^{3}} \cdot\left|\cos \frac{\pi x}{5}\right| \\
& m\left(\sqrt{m}(x-3)^{2}+\frac{1}{\sqrt{m}(x-3)^{2}}\right) \leq m^{\frac{3}{4}}\left|\cos \frac{\pi x}{5}\right| \leq m^{\frac{3}{4}}
\end{aligned}
$$

In the parentheses, we obtained the sum of reciprocals, which have the property $a+\frac{1}{a} \geq 2$.

Consequently, we get the system of inequalities

$$
\left\{\begin{array}{l}
\sqrt{m}(x-3)^{2}+\frac{1}{\sqrt{m}(x-3)^{2}} \geq 2 \\
\sqrt{m}(x-3)^{2}+\frac{1}{\sqrt{m}(x-3)^{2}} \leq \frac{1}{m^{\frac{1}{4}}}
\end{array}\right.
$$

This system has a solution only if $\frac{1}{m^{\frac{1}{4}}}=2, m^{\frac{1}{4}}=\frac{1}{2}, m=\frac{1}{16}=0.0625$.

Answer. 0.0625.