# 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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