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Home » Now compute the sum of all six terms: - Upplift

Now compute the sum of all six terms: - Upplift

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 Another possibility: minimum depth means the depth at which the layer is shallowest over time, but $ d(t) $ could be the rate or amplitude — but the problem says depth ... modeled by a cubic polynomial.

📌 Let’s re-express: perhaps $ d(t) $ is cubic, $ d(1)=1, d(2)=8, d(3)=27, d(4)=64 $, so $ d(t) = t^3 $ is the only cubic fitting. The statement about minimum may be a modeling choice — or perhaps it's a trick: although $ t^3 $ has no minimum, the physical system has a local minimum due to other forces — but mathematically, we must go with the data.

📌 But the problem says achieves its minimum depth exactly once — so it must have one.

📌 Therefore, the only way is if $ d(t) $ is not $ t^3 $, but a different cubic that fits the four points and has a unique minimum. But four points force $ d(t) = t^3 $.

📌 Unless — the values are not $ d(1)=1 $, but rather $ d(1) = a(1)^3 + b(1)^2 + c(1) + d = 1 $, etc. — but that’s the same.

📌 Perhaps d(1) = 1 means the depth is 1 at time 1, which is exactly $ t^3 $.

📌 We are stuck.

📌 But recall: a cubic polynomial has derivative a quadratic, so at most two critical points. It can have one local minimum if the derivative has one root and the second derivative is positive.

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