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.
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- 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) $.
- 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 $
- Right: $ 2a x^2 + 2a y^2 $. So satisfies.
