Find the point of inflection Will there be an inflection point if there is no solution for $x$ when $f ''(x) = 0$? For example,
$$
f(x)=\frac{x^2-x+1}{x-1}
$$
with domain $\mathbb{R}-\{1\}$
Also, is that when $x$ is smaller than $1$, $f(x)$ is concave down?

There is no inflection point if there is no solution for $x$ when $f''(x) = 0$.  For your case, if $x > 1$, then $f''(x) > 0$.  If $x < 1$, then $f''(x) < 0$.
Here is the double-derivative of $\dfrac{x^2 - x + 1}{x - 1}$ and its graph
