Find the point of inflection

Question

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?

Answer 1

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