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why is it necessary to filter the solution - ECD Germany

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 \( w = \frac{16 \pm \sqrt{148}}{6} = \frac{16 \pm 2\sqrt{37}}{6} = \frac{8 \pm \sqrt{37}}{3} \).

📌 Approximiere \( \sqrt{37} \approx 6.08 \), also:

📌 \( w_1 \approx \frac{8 - 6.08}{3} \approx 0.64 \), \( w_2 \approx \frac{8 + 6.08}{3} \approx 4.69 \).

📌 Berechne \( f(0.64) \approx (0.64)^3 - 8(0.64)^2 + 9(0.64) - 18 \approx 0.26 - 3.26 + 5.76 - 18 \approx -15.24 < 0 \),

📌 \( f(4.69) \approx (4.69)^3 - 8(4.69)^2 + 9(4.69) - 18 \approx 102.7 - 175.6 + 42.2 - 18 \approx -49.7 < 0 \),

📌 und \( f(6) = 216 - 288 + 54 - 18 = -36 < 0 \), \( f(5) = 125 - 200 + 45 - 18 = -48 < 0 \), \( f(6.5) \approx 274.6 - 338 + 58.5 - 18 = -24.9 < 0 \), \( f(7) = 343 - 392 + 63 - 18 = 6 > 0 \).

📌 Also liegt eine reelle Wurzel zwischen 6.5 und 7, und da es nur eine positive Extremstelle mit fallendem Verlauf nach oben, und \( f(w) \to \infty \) für \( w \to \infty \), und \( f(w) < 0 \) bei kleinen \( w \), gibt es nur **eine** reelle Wurzel (die andere zwei sind komplex oder negativ).

📌 Prüfe Vorzeichenwechsel: \( f(0) = -18 < 0 \), \( f(1) = 1 - 8 + 9 - 18 = -16 < 0 \), \( f(2) = 8 - 32 + 18 - 18 = -24 < 0 \), \( f(3) = 27 - 72 + 27 - 18 = -36 < 0 \), \( f(4) = 64 - 128 + 36 - 18 = -46 < 0 \), \( f(5) = -48 < 0 \), \( f(6) = -36 < 0 \), \( f(6.7) \approx 302.5 - 360.88 + 60.3 - 18 \approx -16.08 \), \( f(6.9) \approx 328.5 - 383.28 + 62.1 - 18 \approx -0.68 \), \( f(6.95) \approx 334.5 - 388.18 + 62.55 - 18 \approx 0.87 > 0 \).

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