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The derivative with respect to y is −2y+2x, which vanishes at y=x. The second derivative is −2, so this point is the unique global maximizer for real y. This optimizer location differs from the x² returned by the function.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| y_star | Maximizing decision |
| x | Fixed parameter |
How it works
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Original context
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}
return sum
}
// FAthena: F_Athena(x) = argmax_y U(x,y) (Strategy - utility maximization)
func FAthena(x float64) float64 {
// Simplified: find max of utility function U(x,y) = -y² + 2xy
// argmax_y: dU/dy = -2y + 2x = 0 => y* = x
// U(x, x) = -x² + 2x² = x²
return x * x
}
// FApollo: F_Apollo(x) = φ^x + φ^{-x} (Mathematics - golden ratio)
func FApollo(x float64) float64 {
return math.Pow(PHI, x) + math.Pow(PHI, -x)
}
// FHermes: F_Hermes(x) = dx/dt approximation (Communication - rate of change)Exact original notation
argmax_y: dU/dy = -2y + 2x = 0 => y* = x