117. Find the minimum value of the expression

$$
(x-5)(x-1)(x-6)(x-2)+9
$$
S o l u t i o n. We have:

$$
\begin{gathered}
y=(x-5)(x-1)(x-6)(x-2)+9=\left(x^{2}-7 x+10\right) \times \\
\times\left(x^{2}-7 x+6\right)+9=\left(x^{2}-7 x\right)^{2}+16\left(x^{2}-7 x\right)+69= \\
=\left(x^{2}-7 x+8\right)^{2}+5
\end{gathered}
$$

From here, $y_{\text {min }}=5$, when $x^{2}-7 x+8=0$, i.e., when

$$
x=\frac{1}{2}(7 \pm \sqrt{17})
$$

A n s w e r: 5 .