Find the open intervals where $f (z) = z^{3} + 6 z^{2} + z + 1$ is increasing and Where it is decreasing.
Enter your answer in interval notation. For example, if the function is increasing when $- 10 < z < 2$ as well as when $4 < z < 7$ , then you would enter $(- 10, 2) \cup (4, 7)$ .
$f (z) = z^{3} + 6 z^{2} + z + 1$

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Determine increasing/decreasing intervals: $(- \infty, - 2 - \frac{\sqrt{33}}{3}) \cup (- 2 + \frac{\sqrt{33}}{3}, \infty)$ for increasing, $(- 2 - \frac{\sqrt{33}}{3}, - 2 + \frac{\sqrt{33}}{3})$ for decreasing

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Step 1 ：Find the first derivative: $f^{'} (z) = 3 z^{2} + 12 z + 1$

Step 2 ：Find the critical points: $z = - 2 - \frac{\sqrt{33}}{3}, - 2 + \frac{\sqrt{33}}{3}$

Step 3 ：Analyze the intervals: $(- \infty, - 2 - \frac{\sqrt{33}}{3})$ , $(- 2 - \frac{\sqrt{33}}{3}, - 2 + \frac{\sqrt{33}}{3})$ , $(- 2 + \frac{\sqrt{33}}{3}, \infty)$

Step 4 ：Determine increasing/decreasing intervals: $(- \infty, - 2 - \frac{\sqrt{33}}{3}) \cup (- 2 + \frac{\sqrt{33}}{3}, \infty)$ for increasing, $(- 2 - \frac{\sqrt{33}}{3}, - 2 + \frac{\sqrt{33}}{3})$ for decreasing

Find the open intervals where f(z)=z3+6z2+z+1 is increasing and Where it is decreasing.Enter your answer in interval notation. For example, if the function is increasing when −10<z<2 as well as when 4<z<7, then you would enter (−10,2)∪(4,7).f(z)=z3+6z2+z+1

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