Provide the smallest possible interval containing the roots of

$$
x^{2014}-100 x+1
$$
If $x \leqslant 0$ then the expression is $\geqslant 1$, so $x$ is not a root.

If $x \geqslant 100^{1 / 2013}$, then $x^{2014} \geqslant 100 x > 100 x - 1$.

If $0 < x < 100^{1 / 2013}$, then $x^{2014} + 1 - 100 x \geqslant 0$.

Therefore, the roots are within $\left[\frac{1}{100}, 100^{1 / 2013}\right]$.