Let's determine the values of the coefficients in the function $y=x^{3}+a x^{2}+b x+c$ such that the coordinates of the extreme values and the inflection point are integers!
$y^{\prime}=3 x^{2}+2 a x+b ; y^{\prime \prime}=6 x+2 a$. From this, the abscissa of the inflection point is $x_{i}=-\frac{a}{3}$. This is an integer if $a=3 n$. The abscissae of the extreme values are:

$$
x_{m}=\frac{-6 n \pm \sqrt{36 n^{2}-12 b}}{6}=-n \pm \sqrt{n^{2}-\frac{b}{3}}
$$

so $\sqrt{n^{2}-\frac{b}{3}}=p$, which must be an integer. From this, $b=3\left(n^{2}-p^{2}\right)$. Finally, for the ordinate to also be an integer at these points, $c$ must also be an integer. Thus, expressions of the form $y=x^{3}+3 n x^{2}+3\left(n^{2}-p^{2}\right) x+c$ satisfy our conditions, where $n, p, c$ are integers.