Suppose $f(x) = g(x) h(x)$ and the number of direction changes in both $g$ and $h$ are known. Is there a simple way to prove that the number of direction changes in $f$ is bounded by the number of direction changes of $g$ and $h$?
As a simple case, consider $h$ as monotonic - it seems intuitive that $f$ cannot have more direction changes than does $g$.