jmc

We can think of $|x+15|$ as the distance between $x$ and $-15$ on the real numbers line, and $|x-19|$ as the distance between $x$ and 19 on the real number line.



By the Triangle Inequality, the sum of these distances is at least $19-(-15)=34,$ which implies that at least one of $|x+15|$ and $|x-19|$ is always at least 17. Therefore, $f(x)\ge 17.$

Note that $f(2)=17,$ so the minimum value of $f(x)$ is $\boxed{17}.$