Example 5 A unit spent 500,000 yuan to purchase a piece of high-tech equipment. According to the tracking survey of this model of equipment, after the equipment is put into use, if the maintenance and repair costs are averaged to each day, the conclusion is: the maintenance and repair cost on the $x$-th day is $\left[\frac{1}{4}(x-1)+500\right]$ yuan.
Solution: (1) If the equipment is put into use for $x$ days, and the average daily loss is $y$ yuan, then
$$
\begin{aligned}
y= & \left[500000+\left(\frac{1}{4} \times 0+500\right)+\left(\frac{1}{4} \times 1+500\right)+\right. \\
& \left.\cdots+\left(\frac{x-1}{4}+500\right)\right] \div x \\
= & \frac{1}{x}\left[500000+500 x+\frac{1}{4} \cdot \frac{x(x-1)}{2}\right] \\
= & \frac{500000}{x}+\frac{x}{8}+499 \frac{7}{8} .
\end{aligned}
$$
$$
\begin{array}{l}
\text { (2) } y=\frac{500000}{x}+\frac{x}{8}+499 \frac{7}{8} \\
\geqslant 2 \sqrt{\frac{500000}{x} \cdot \frac{x}{8}}+499 \frac{7}{8} \\
=500+499 \frac{7}{8}=999 \frac{7}{8} .
\end{array}
$$

Equality holds if and only if $\frac{500000}{x}=\frac{x}{8}$, i.e., $x=2000$,
at which point, $y$ reaches its minimum value.
Therefore, the equipment should be scrapped after being put into use for 2000 days.