Find the minimum value of the expression

$$
\sqrt{1-x+x^{2}}+\sqrt{1-\sqrt{3} \cdot x+x^{2}}
$$
Solution. The expression is defined for all real $x$, since

\[
\begin{gathered}
\sqrt{1-x+x^{2}}+\sqrt{1-\sqrt{3} \cdot x+x^{2}}= \\
=\sqrt{\left(x-\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}+\sqrt{\left(x-\frac{\sqrt{3}}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}}
\end{gathered}
\]

Let the coordinates of point $A$ in the Cartesian coordinate system be $\left(\frac{1}{2} ; \frac{\sqrt{3}}{2}\right)$, the coordinates of point $B$ be $\left(\frac{\sqrt{3}}{2} ;-\frac{1}{2}\right)$, and the coordinates of point $C$ be $(x ; 0)$. Points $A$ and $B$ are fixed, while point $C$ is any point on the $x$-axis. Then

\[
A C=\sqrt{\left(x-\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}, \quad B C=\sqrt{\left(x-\frac{\sqrt{3}}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}}
\]

The question is, at which point on the $x$-axis will the length of $A C + B C$ be minimal. If point $C$ is the intersection of segment $A B$ and the $x$-axis, then the sum of the two distances is exactly the length of segment $A B$. For any other point $C'$, the triangle inequality holds:

\[
A C^{\prime}+B C^{\prime}>A B
\]

The sought minimum is the distance $A B$:

\[
\begin{aligned}
A B & =\sqrt{\left(\frac{1}{2}-\frac{\sqrt{3}}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}-\left(-\frac{1}{2}\right)\right)^{2}}= \\
& =\sqrt{2 \cdot\left(\frac{\sqrt{3}}{2}\right)^{2}+2 \cdot\left(\frac{1}{2}\right)^{2}}=\sqrt{\frac{3}{2}+\frac{1}{2}}=\sqrt{2}
\end{aligned}
\]

This minimum value is obtained at the point where line $A B$ intersects the $x$-axis.

The equation of line $A B$ is

\[
\begin{gathered}
\frac{\frac{\sqrt{3}}{2}+\frac{1}{2}}{\frac{1}{2}-\frac{\sqrt{3}}{2}} \cdot x+\frac{\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}-\frac{1}{2} \cdot\left(-\frac{1}{2}\right)}{\frac{\sqrt{3}}{2}-\frac{1}{2}}=y \\
y=\frac{\sqrt{3}+1}{1-\sqrt{3}} \cdot x+\frac{2}{\sqrt{3}-1}
\end{gathered}
\]

The intersection with the $x$-axis will be the point whose second coordinate is 0.

\[
x=\frac{2}{\sqrt{3}+1}=\sqrt{3}-1
\]

Thus, the minimum of the expression $\sqrt{1-x+x^{2}}+\sqrt{1-\sqrt{3} \cdot x+x^{2}}$ is $\sqrt{2}$, and the point of minimum is $x=\sqrt{3}-1$.