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Home » \( \frac(x - 2)(x + 2)x - 2 \) - Network4

\( \frac(x - 2)(x + 2)x - 2 \) - Network4

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 Alternatively, known result: For fixed perimeter, the right triangle of maximum area is isosceles. But to find maximum area algebraically, use calculus or known identity.

📌 But simpler: among all right triangles with fixed perimeter, maximum area occurs when $ a = b $. So use that.

📌 Question: Let $ f(x) $ be a cubic polynomial such that $ f(1) = 10 $, $ f(2) = 20 $, $ f(3) = 30 $, and $ f(4) = 40 $. Find $ f(5) $.

📌 Let $ f(x) $ be a cubic polynomial. Define $ g(x) = f(x) - 10x $. Then:

📌 So $ g(x) $ has roots at $ x = 1, 2, 3, 4 $. But $ g(x) $ is a cubic polynomial, while a cubic polynomial cannot have 4 distinct roots unless it is identically zero — contradiction.

📌 Therefore, such a cubic polynomial cannot exist — unless the values force a special form. But the values $ f(1)=10, f(2)=20, f(3)=30, f(4)=40 $ suggest $ f(x) - 10x $ has four zeros. So define:

📌 Then $ g(1) = g(2) = g(3) = g(4) = 0 $. So $ g(x) $ has roots at $ 1,2,3,4 $, so $ g(x) = k(x-1)(x-2)(x-3)(x-4) $, but this is degree 4 — contradiction, since $ f(x) $ is cubic.

📌 Unless — the only way this is possible is if $ g(x) $ is not degree 4 — but it must be degree 4 unless $ f(x) $ is not truly cubic. But the problem says it is cubic.

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