$6 \cdot 78$ Let $f(x)=|x-p|+|x-15|+|x-p-15|$, where $0<p<15$. Try to find the minimum value of $f(x)$ for $x$ in the interval $p \leqslant x \leqslant 15$.
[Solution] From the problem, we know that $0 < p < 15$.

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