Example 2 Given that the volume of a rectangular prism is 1, the sum of its length, width, and height is $k$, and its surface area is $2k$. Find the range of the real number $k$.


【Analysis】Let the length, width, and height of the rectangular prism be $a$, $b$, and $c$ respectively. Then
$$
a b c=1, a+b+c=k, a b+b c+c a=k .
$$

According to Vieta's formulas, $a$, $b$, and $c$ are the three roots of the equation
$$
x^{3}-k x^{2}+k x-1=0
$$

We have
$$
\begin{array}{l}
\text { Also, } x^{3}-k x^{2}+k x-1 \\
=(x-1)\left[x^{2}+(1-k) x+1\right],
\end{array}
$$

Without loss of generality, let $a=1$. Then $b$ and $c$ are the two roots of the equation
$$
x^{2}+(1-k) x+1=0
$$

Thus, we have
$$
\text { Therefore, }\left\{\begin{array}{l}
\Delta=(1-k)^{2}-4 \geqslant 0, \\
b+c=k-1>0, \\
b c=1>0 .
\end{array}\right.
$$

Solving this, we get $k \geqslant 3$.
Therefore, the range of the real number $k$ is $[3,+\infty)$.