\(f(x)=ax^3+bx^2+cx+d\) , \(f' (1)=f' (-1)=0\) and \(a>0\)

The region bounded by \(x\)-axis and \(f(x) \) are \(''A_1,A_2,A_3''\) such that \(A1\) and \(A3\) are in \(IV\) and \(II\) quadrant respectively.

If \(f(x) \) has roots \(a_1,a_2,a_3\), such that \(a_1>a_2\) and \(a1>0\) Also \((a_1,a_2,a_3)\), \((a_1^2,a_2^2,a_3^2)\), \((a_1^4,a_2^4,a_3^4)\) , each triplets are in an arithmetic progression.

Then which of the following must be true?

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