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But $ d(t) = t^3 $ has $ d'(t) = 3t^2 $, which is zero only at $ t=0 $, but $ d''(t) = 6t $, so $ d''(0) = 0 $ — inconclusive.
So not a strict minimum.
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.

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