To find the extrema of the function $f(x) = \sqrt[3]{x^3 - 6x^2}$, we will follow these steps:

1. Find the derivative $f'(x)$.
2. Set $f'(x) = 0$ to find critical points.
3. Analyze the critical points and intervals using the first or second derivative test to classify extrema.

Let’s calculate step by step.

Step 1: Derivative of $f(x)$

The derivative of a cube root function $\sqrt[3]{g(x)}$ is: $\frac{d}{dx} \sqrt[3]{g(x)} = \frac{1}{3} g(x)^{-2/3} \cdot g'(x)$ Here, $g(x) = x^3 - 6x^2$ and $g'(x) = 3x^2 - 12x$.

So, $f'(x) = \frac{1}{3} (x^3 - 6x^2)^{-2/3} \cdot (3x^2 - 12x)$

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

Step 2: Solve $f'(x) = 0$

The numerator $3x^2 - 12x$ must equal zero: $3x(x - 4) = 0$ Thus, $x = 0$ or $x = 4$.

The denominator $(x^3 - 6x^2)^{2/3} \neq 0$ for these points since it only vanishes at $x = 0$ or $x = 6$, but cube roots are defined everywhere.

Step 3: Analyze Critical Points

We will examine $x = 0$ and $x = 4$ using the first derivative test.

First Derivative Test

• For $x < 0$, choose $x = -1$: $f'(x) > 0$ (positive slope).
• For $0 < x < 4$, choose $x = 2$: $f'(x) < 0$ (negative slope).
• For $x > 4$, choose $x = 5$: $f'(x) > 0$ (positive slope).

Behavior:

• At $x = 0$, $f(x)$ changes from increasing to decreasing. This is a local maximum.
• At $x = 4$, $f(x)$ changes from decreasing to increasing. This is a local minimum.

Extrema:

• Local maximum at $x = 0$, $f(0) = \sqrt[3]{0^3 - 6 \cdot 0^2} = 0$.
• Local minimum at $x = 4$, $f(4) = \sqrt[3]{4^3 - 6 \cdot 4^2} = \sqrt[3]{64 - 96} = \sqrt[3]{-32} = -2$.

Final Answer:

• Local maximum: $(0, 0)$
• Local minimum: $(4, -2)$

Would you like further details or explanations?

---

Related Questions:

1. How do you find inflection points for this function?
2. What is the second derivative $f''(x)$ for further analysis?
3. How do we graph $f(x) = \sqrt[3]{x^3 - 6x^2}$?
4. What are the domain and range of $f(x)$?
5. How does the behavior of $f(x)$ change as $x \to \pm\infty$?

Tip:

When dealing with cube roots, always pay attention to the behavior near zero, as the derivative can have unique properties.