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Home » Now evaluate \( R(-2) \): - Portal da AcÃºstica

Now evaluate \( R(-2) \): - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Thus, $ f(x) = x^2 + 2x $, a quadratic, but consistent with the data. Despite being expected cubic, the data fits quadratic. Since the third differences are zero (first: 5,7,9; second: 2,2), third differences are zero, confirming degree â¤ 2.

📌 So $ f(5) = 5^2 + 2\cdot5 = 25 + 10 = 35 $.

📌 Even though the problem speculates cubic, the data fits a quadratic exactly. The cubic coefficient is zero.

📌 Therefore, the reconstructed value is $ oxed{35} $.

📌 Question: A brain-computer interface algorithm uses a function $ f $ satisfying $ f(x + y) + f(x - y) = 2f(x) + 2f(y) $ for all real $ x, y $. If $ f(1) = 3 $, compute $ f(5) $.

📌 Solution: The functional equation is $ f(x+y) + f(x-y) = 2f(x) + 2f(y) $. This is a known variant of the quadratic functional equation.

📌 Assume $ f(x) = ax^2 $. Test:

📌 Left: $ a(x+y)^2 + a(x-y)^2 = a(x^2 + 2xy + y^2 + x^2 - 2xy + y^2) = a(2x^2 + 2y^2) = 2a x^2 + 2a y^2 $

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