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Lagrange and Taylor on the same polynomial. Let \(P \in \mathbb{R}_2[X]\) (degree at most \(2\)) be the polynomial determined by \(P(0) = 1\), \(P(1) = 2\), \(P(2) = 5\).
  1. Compute \(P\) explicitly via the Lagrange formula at the nodes \(x_0 = 0\), \(x_1 = 1\), \(x_2 = 2\).
  2. Compute \(P(1)\), \(P'(1)\), \(P''(1)\).
  3. Verify that the Taylor expansion of \(P\) at \(a = 1\) recovers the same polynomial: $$ P(X) = P(1) + P'(1)(X - 1) + \tfrac{P''(1)}{2!}(X - 1)^2. $$

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