10,11  [ Methods for solving problems with parameters ]

For all values of the parameter $a$, find the number of real roots of the equation $x^{3}-x-a=0$.

#
Setting the derivative $3 x^{2}-1$ to zero, we find that the function $x^{3}-x$ has a maximum of $\frac{2 \sqrt{3}}{9}$ at the point $x_{1}=-\frac{\sqrt{3}}{3}$ and a minimum of $-\frac{2 \sqrt{3}}{9}$ at the point $x_{2}=\frac{\sqrt{3}}{3}$. From this, we get the answer.

## Answer

Three roots when $|a|<\frac{2 \sqrt{3}}{9}$, two roots, one of which is a double root, when $|a|=\frac{2 \sqrt{3}}{9}$.