Answer: 15

Small Hint:

Determine the sign of each expression inside an absolute value on the given interval

Big Hint:

On $p\leq x\leq15,$ simplify $f(x)$ to a decreasing linear function

Solution:

Since $p\leq x\leq15,$ we have $$\begin{aligned} |x-p|&=x-p,\\ |x-15|&=15-x,\\ |x-p-15|&=p+15-x. \end{aligned}$$ Hence $f(x)=30-x.$ This is minimized at the right endpoint $x=15,$ where its value is $15.$