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 $(g(x))^{2}+$ $5 f(x)$, if the minimum value of the function $(f(x))^{2}+5 g(x)$ is -17.
Answer: $\frac{9}{2}$.

Solution. Let $f(x)=a x+b, g(x)=a x+c$, where $a \neq 0$. Consider $h(x)=(f(x))^{2}+5 g(x)$. Expanding the brackets, we get $h(x)=(a x+b)^{2}+5(a x+c)=a^{2} x^{2}+a(2 b+5) x+b^{2}+5 c$. 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 $x_{\mathrm{B}}=-\frac{2 b+5}{2 a}$; the y-coordinate of the vertex is $h\left(x_{\mathrm{B}}\right)=-5 b-\frac{25}{4}+5 c$.

Similarly, we obtain that the minimum value of the expression $(g(x))^{2}+5 f(x)$ is $-5 c-\frac{25}{4}+5 b$. Note that the sum of these two minimum values is $-\frac{25}{2}$, therefore, if one of these minimum values is -17, then the other is $-\frac{25}{2}+17=\frac{9}{2}$.