3. What is the smallest area that a circle centered at the origin can have, which has common points with the graph of the function $y=\frac{6}{x}-\frac{4 x}{3}$?

(16 points)
Solution. $(x ; f(x))$ belongs to a circle with radius $r$

$r^{2}=f^{2}(x)+x^{2}=\left(\frac{6}{x}-\frac{4 x}{3}\right)^{2}+x^{2}=\frac{36}{x^{2}}+\frac{25 x^{2}}{9}-16\left(r^{2}\right)^{\prime}=-\frac{72}{x^{3}}+\frac{50 x}{9}=\frac{2\left(25 x^{4}-36 \cdot 9\right)}{16 x^{3}}$,

$x_{\min }=\sqrt{18 / 5}, \quad S_{\min }=\pi r_{\min }^{2}=4 \pi$.

Answer: $4 \pi$.