The function given in the image is:

$f(x) = \sqrt[3]{x^3 - 6x^2}$

To find the intervals of monotonicity, we need to follow these steps:

Step 1: Compute the derivative

The derivative of $f(x)$ can be found using the chain rule. Rewrite the function as: $f(x) = (x^3 - 6x^2)^{1/3}$

The derivative is: $f'(x) = \frac{1}{3}(x^3 - 6x^2)^{-2/3} \cdot (3x^2 - 12x)$

Simplify the derivative: $f'(x) = \frac{(3x^2 - 12x)}{3(x^3 - 6x^2)^{2/3}}$ $f'(x) = \frac{x(x - 4)}{(x^3 - 6x^2)^{2/3}}$

Step 2: Critical points

The critical points occur where $f'(x) = 0$ or $f'(x)$ is undefined.

1. Numerator $x(x - 4) = 0$: $x = 0 \quad \text{or} \quad x = 4$

2. Denominator $(x^3 - 6x^2)^{2/3} = 0$: $x^3 - 6x^2 = 0 \implies x^2(x - 6) = 0$ $x = 0 \quad \text{or} \quad x = 6$

Thus, the critical points are $x = 0, 4, 6$.

Step 3: Test intervals

We analyze the sign of $f'(x)$ in the intervals determined by the critical points: $(-\infty, 0)$, $(0, 4)$, $(4, 6)$, and $(6, \infty)$.

• Interval $(-\infty, 0)$: Choose $x = -1$. $f'(-1) = \frac{(-1)(-1 - 4)}{((-1)^3 - 6(-1)^2)^{2/3}} = \frac{(-1)(-5)}{(-1 - 6)^{2/3}} > 0$

  $f'(x) > 0$, so $f(x)$ is increasing.

• Interval $(0, 4)$: Choose $x = 1$. $f'(1) = \frac{(1)(1 - 4)}{((1)^3 - 6(1)^2)^{2/3}} = \frac{(1)(-3)}{(1 - 6)^{2/3}} < 0$

  $f'(x) < 0$, so $f(x)$ is decreasing.

• Interval $(4, 6)$: Choose $x = 5$. $f'(5) = \frac{(5)(5 - 4)}{((5)^3 - 6(5)^2)^{2/3}} = \frac{(5)(1)}{(125 - 150)^{2/3}} < 0$

  $f'(x) < 0$, so $f(x)$ is decreasing.

• Interval $(6, \infty)$: Choose $x = 7$. $f'(7) = \frac{(7)(7 - 4)}{((7)^3 - 6(7)^2)^{2/3}} = \frac{(7)(3)}{(343 - 294)^{2/3}} > 0$

  $f'(x) > 0$, so $f(x)$ is increasing.

Step 4: Intervals of monotonicity

• $f(x)$ is increasing on $(-\infty, 0) \cup (6, \infty)$.
• $f(x)$ is decreasing on $(0, 6)$.

Would you like more details or help with another aspect of this problem?

Here are 5 related questions to expand your understanding:

1. How do you find concavity and points of inflection for a function like this?
2. What is the significance of critical points in the behavior of a function?
3. How can you sketch the graph of $f(x)$ using this information?
4. How would the monotonicity change if $-6x^2$ were replaced by $+6x^2$?
5. How do you determine the behavior of $f(x)$ near points where the derivative is undefined?

Tip: Always rewrite the function in simpler forms when taking derivatives—it helps avoid mistakes with the chain rule or exponent rules!