When all's unknown

Calculus Level 5

\[ \large{ f(x) = ax^3 + bx^2 + cx + d} \]

The Taylor Series expansion of \( f(x) \) about \( x= a\) is

\[\frac{d}{6}(x- a)^3 + (3a + b + 2d)(x- a)^2 + (3a^2 + c + 6d)(x - a) \]

Determine the product \(abcd\), where \(a\), \(b\), \(c\), and \(d\) are distinct, nonzero, coprime (not necessarily pairwise) integers.

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