Find the intervals in which the function
$f(x) = -2x^3 - 9x^2 - 12x + 1$
is strictly increasing or strictly decreasing.

Q: Find the intervals in which the function $f(x) = -2x^3 - 9x^2 - 12x + 1$ is strictly increasing or strictly decreasing.

Explanation: We have: \[f'(x) = -6x^2 - 18x - 12 = -6(x + 1)(x + 2)\] Setting $f'(x) = 0$: \[x = -1,\ -2\] Intervals: $(-\infty, -2),\ (-2, -1),\ (-1, \infty)$ \[\] In $(-\infty, -2)$ and $(-1, \infty)$, $f'(x) 0$ ⟹ strictly increasing

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Accepted Answer

Explanation: We have:
$$f'(x) = -6x^2 - 18x - 12 = -6(x + 1)(x + 2)$$

Setting $f'(x) = 0$:
$$x = -1,\ -2$$

Intervals: $(-\infty, -2),\ (-2, -1),\ (-1, \infty)$

In $(-\infty, -2)$ and $(-1, \infty)$, $f'(x) < 0$ ⟹ strictly decreasing

In $(-2, -1)$, $f'(x) > 0$ ⟹ strictly increasing

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