I. (25 points) The Xiguang Factory glasses workshop has received a batch of tasks, requiring the processing of 6000 Type A parts and 2000 Type B parts. This workshop has 214 workers, and each of them can process 3 Type B parts in the time it takes to process 5 Type A parts. These workers are to be divided into two groups, with both groups working simultaneously, each group processing one type of part. To complete this batch of tasks in the shortest time, how should the workers be divided?
One, Solution: Let the number of people in the group that includes those with the surname $A$ be $x$, and in a unit of time, the number of $\mathrm{C} . \mathrm{A}$ parts processed by one person is $5 k$, then the number of the other parts is $3 k$.
The time required to process type $A$ parts is $t_{A}(x)=\frac{6000}{5 k x}$;
The time required to process type $B$ parts is $t_{B}(x)=$ $\frac{2000}{3 k(214-x)}$.
$\therefore$ The total time to complete the entire task is
$$
t(x)=\max \left\{t_{A}(x), t_{B}(x)\right\}=\frac{2000}{k} \cdot f(x),
$$

where, $f(x)=\max \left\{\frac{3}{5 x}, \frac{1}{3(214-x)}\right\}$.
Thus, the problem is converted to finding the natural number $x(1 \leqslant x \leqslant 213)$, such that the function $f(x)$ takes the minimum value.
In the interval $[1,213]$, $\frac{3}{5 x}$ is a decreasing function, and $\frac{1}{3(214-x)}$ is an increasing function, so the minimum value of $f(x)$ is obtained at $x_{0}=137 \frac{4}{7}$, where $x_{0}$ satisfies the equation $\frac{3}{5 x}=\frac{1}{3(214-x)}$.
$\because x_{0}=137 \frac{4}{7}$ is not an integer, and $f(137)<f(138)$,
$\therefore$ The number of people in the group processing type $A$ parts is 137, and the number of people in the other group is 77.