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(A) point of inflection occurs where the second derivative $f''(x) = 0$ and changes sign.First,find the first derivative: $f'(x) = a x^2 + 2(a + 2)x +

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$(-\infty, -2) \cup (0, \infty)$

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(A) point of inflection occurs where the second derivative $f''(x) = 0$ and changes sign.First,find the first derivative: $f'(x) = a x^2 + 2(a + 2)x + (a - 1)$.Next,find the second derivative: $f''(x) = 2ax + 2(a + 2)$.Set $f''(x) = 0$ to find the point of inflection: $2ax + 2(a + 2) = 0 \implies x = -\frac{a + 2}{a}$.For the point of inflection to be negative,we require $x  0$.Solving the inequality $\frac{a + 2}{a} > 0$ using the sign scheme method,we find the critical points at $a = -2$ and $a = 0$.The expression is positive in the intervals $(-\infty, -2)$ and $(0, \infty)$.Thus,the set of values for $a$ is $(-\infty, -2) \cup (0, \infty)$.

The set of value$(s)$ of $a$ for which the function $f(x) = \frac{a x^3}{3} + (a + 2) x^2 + (a - 1) x + 2$ possesses a negative point of inflection.

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