# Problem 1.

A numerical characteristic $x$ of a certain thermal power process is a root of the equation

$$
x^{3}-3 x=t
$$

where $t$ is the ambient temperature measured in degrees Celsius. For some technological reasons, the root must be unique. For what values of $t$ does the equation have a unique root $x_{0}$? Estimate the absolute value of this root from below and show that the obtained estimate cannot be improved.
# Solution.

It is not difficult to plot the graph of the function $y=x^{3}-3 x$, noting that this function is odd, equals zero at exactly three points $x=0, \pm \sqrt{3}$, $(1; -2)$ is a point of minimum, and $(-1; 2)$ is a point of maximum. The function increases without bound for $x>1$ and decreases without bound for $x>2$.

The only root exists precisely when $|t|>2$.

Next, we estimate the absolute value of the root $x$ when $|t|>2$. From the graph, it is clear that $|x|>\sqrt{3}$. A more precise estimate can be obtained by considering the inequalities $x^{3}-3 x>2$ for $t>2$ and $x^{3}-3 x<2$ for $t<2$.

A geometric solution is also acceptable, based on the observation that the limiting (intermediate) case of two roots corresponds to the situation where one of the roots is a point of extremum.

## Answer.

The equation has a unique root precisely when $|t|>2$. For this root, the inequality $|x|>2$ holds.