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 $2(g(x))^{2}-$ $f(x)$, if the minimum value of the function $2(f(x))^{2}-g(x)$ is $\frac{7}{2}$.
Answer: $-\frac{15}{4}$.

Solution. Let $f(x)=a x+b, g(x)=a x+c$, where $a \neq 0$. Consider $h(x)=2(f(x))^{2}-g(x)$. Expanding the brackets, we get $h(x)=2(a x+b)^{2}-(a x+c)=2 a^{2} x^{2}+a(4 b-1) x+2 b^{2}-c$. 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 $x_{\mathrm{B}}=-\frac{4 b-1}{4 a}$; the y-coordinate of the vertex is $h\left(x_{\mathrm{B}}\right)=b-\frac{1}{8}-c$.

Similarly, we obtain that the minimum value of the expression $2(g(x))^{2}-f(x)$ is $c-\frac{1}{8}-b$. Notice that the sum of these two minimum values is $-\frac{1}{4}$, therefore, if one of these minimum values is $\frac{7}{2}$, then the other is $-\frac{1}{4}-\frac{7}{2}=-\frac{15}{4}$.