$\begin{array}{c}6 \cdot 78 \text { Let } f(x)=|x-p|+|x-15|+|x-p-15| \text {, where } \\ 0<p<15 \text {. Find the minimum value of } f(x) \text { for } x \text { in the interval } p \leqslant x \leqslant 15. \end{array}$
[Solution] From the given, we know $0 < p < 15$.

Therefore, $f(x) = (x - p) + (15 - x) + (p + 15 - x)$, which simplifies to $f(x) = 30 - x$.
The minimum value of this function on the interval $[p, 15]$ is $f(15) = 15$.