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Calculus Level 4

If the curve \(f(x)=2{ x }^{ 3 }+a{ x }^{ 2 }+bx\), where \(a\) and \(b\) are positive integers, cuts the \(x\)-axis at three distinct points. Then find the minimum value of \(a +b\).

Bonus: Also find all \(f(x)\) for which \(a+b\) is minimum.

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