Bounded and cubic

Algebra Level 4

Consider a general cubic polynomial function \(f(x) = ax^3 + bx^2 + cx + d\) with real constants \(a>0 , b>0, c\ne 0, d \ne 0 \) such that for all \(|x| \leq 1 \), the inequality \( |f(x) | \leq 1 \) is fulfilled.

Find \(\max \left(a + b + |c| + |d| \right)\).

\[\] Notation: \( | \cdot | \) denotes the absolute value function.

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