But suppose the oceanographer observes that depth decreases then increases, with a single minimum at some $ t = m > 0 $. But our interpolation forces $ d(t) = t^3 $, which decreases then increases? No — $ t^3 $ increases for $ t > 0 $, $ d'(t) = 3t^2 \geq 0 $, so it is increasing everywhere for $ t > 0 $, never decreases. - Appfinity Technologies