2. Given two linear functions $f(x)$ and $g(x)$ such that the graphs $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. Find the minimum value of the function $3(g(x))^{2}+$ $2 f(x)$, if the minimum value of the function $3(f(x))^{2}+2 g(x)$ is $-\frac{19}{6}$.
Answer: $\frac{5}{2}$.

Solution. Let $f(x)=a x+b, g(x)=a x+c$, where $a \neq 0$. Consider $h(x)=3(f(x))^{2}+$ $2 g(x)$. Expanding the brackets, we get $h(x)=3(a x+b)^{2}+2(a x+c)=3 a^{2} x^{2}+2 a(3 b+1) x+3 b^{2}+$ 2c. The graph of $y=h(x)$ is a parabola opening upwards, and the minimum value is attained at the vertex. The x-coordinate of the vertex is $x_{\mathrm{v}}=-\frac{3 b+1}{3 a}$; the y-coordinate of the vertex is $h\left(x_{\mathrm{v}}\right)=-2 b-$ $\frac{1}{3}+2 c$.

Similarly, we obtain that the minimum value of the expression $3(g(x))^{2}+2 f(x)$ is $-2 c-\frac{1}{3}+2 b$. Note that the sum of these two minimum values is $-\frac{2}{3}$, therefore, if one of these minimum values is $-\frac{19}{6}$, then the other is $-\frac{2}{3}+\frac{19}{6}=\frac{5}{2}$.