Let the polynomial $P (x)$ with integer coefficients have a local minimum at the point $x = \sqrt 2$. Prove that $P (x)$ has also a local minimum or maximum at the point $x = -\sqrt 2$ (not an inflection point).
Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$, where $a_i \in \mathbb Z, i=0,..n$. Then $$P'(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+...+a_1$$ $$P'(\sqrt2)=0$$
For a polynomial $Q$ with integer coefficients, $Q(\sqrt 2)=0$ iff $X^2-2$ divides $Q(X)$ iff $Q(-\sqrt 2)=0$.
That $P$ has an extremum at $\sqrt 2$ is equivalent to $P'(\sqrt 2)=P''(\sqrt 2)=\ldots=P^{(m)}(\sqrt 2)=0$ and $P^{(m+1)}(\sqrt 2)\ne 0$ for some odd number $m$, and by the above remark, thius again is equivalent to $P'(-\sqrt 2)=P''(-\sqrt 2)=\ldots=P^{(m)}(-\sqrt 2)=0$ and $P^{(m+1)}(-\sqrt 2)\ne 0$, i.e., that $P$ has an extremum at $-\sqrt 2$.
