171. Maximum - minimum without analysis. Find the maximum and minimum values of the expression $\frac{x^{2}-2 x+2}{2 x-2}$, without using methods of mathematical analysis.
171. Let's perform the transformations:

$$
f(x)=\frac{x^{2}-2 x+2}{2 x-2}=\frac{1}{2}\left[x-1+\frac{1}{x-1}\right]=\frac{x^{2}}{2(x-1)}-1
$$

The sum of two reciprocals takes the smallest absolute value when these numbers are equal to $\pm 1$. Therefore, $f(x)$ takes the smallest absolute value

![](https://cdn.mathpix.com/cropped/2024_05_21_2b40d984baf9da275c68g-131.jpg?height=594&width=616&top_left_y=317&top_left_x=720)

when $x-1= \pm 1$, that is, when $x=2$ or 0. From the last expression for $f(x)$, it becomes clear that $f(x)$ is not bounded above for $x>2$, not bounded below for $x<0$, and is undefined at $x=1$.

From this, it follows that at the point $x=2$, $f(x)$ has a local minimum, equal to 1, and at the point $x=0$, it has a local maximum, equal to -1.