(c) Find the critical points of $$ f(x)=x^{1 / 3}(x-4) $$. Sketch the graph of $$ f(x) $$.

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To find the **critical points** of the function $$ f(x) = x^{1/3}(x - 4) $$, we'll follow these steps:

### 1. Find the Derivative $$ f'(x) $$

First, let's differentiate $$ f(x) $$ with respect to $$ x $$:

$$
f(x) = x^{1/3}(x - 4) = x^{4/3} - 4x^{1/3}
$$

Differentiating term by term:

$$
f'(x) = \frac{4}{3}x^{1/3} - \frac{4}{3}x^{-2/3}
$$

### 2. Determine Where $$ f'(x) = 0 $$ or $$ f'(x) $$ Does Not Exist

**a. Setting $$ f'(x) = 0 $$:**

$$
\frac{4}{3}x^{1/3} - \frac{4}{3}x^{-2/3} = 0
$$

Multiply both sides by $$ \frac{3}{4}x^{2/3} $$ to eliminate the fractions and negative exponents:

$$
x^{1/3} \cdot x^{2/3} - 1 = 0 \implies x - 1 = 0 \implies x = 1
$$

**b. Checking Where $$ f'(x) $$ Does Not Exist:**

The derivative $$ f'(x) $$ involves the term $$ x^{-2/3} $$, which is undefined at $$ x = 0 $$. Therefore, $$ x = 0 $$ is also a critical point.

### 3. Identify the Critical Points

- **At $$ x = 1 $$:**
  $$
  f(1) = 1^{1/3}(1 - 4) = 1 \times (-3) = -3
  $$
  So, one critical point is $$ (1, -3) $$.

- **At $$ x = 0 $$:**
  $$
  f(0) = 0^{1/3}(0 - 4) = 0 \times (-4) = 0
  $$
  Another critical point is $$ (0, 0) $$.

### 4. Summary of Critical Points

- $$ (0, 0) $$
- $$ (1, -3) $$

### 5. Sketch of the Graph of $$ f(x) $$

Here's a rough sketch of the function $$ f(x) = x^{1/3}(x - 4) $$:

![Graph of f(x) = x^(1/3)(x - 4)](https://i.imgur.com/your_image_link.png)

**Key Features to Include in the Sketch:**

1. **Critical Points:**
   - Point at $$ (0, 0) $$: This is where the derivative does not exist.
   - Point at $$ (1, -3) $$: This is where the derivative is zero.

2. **Behavior Around Critical Points:**
   - **At $$ x = 0 $$:** The function changes its slope direction, indicating a possible inflection point.
   - **At $$ x = 1 $$:** This is likely a local minimum since the function decreases before $$ x = 1 $$ and increases after.

3. **End Behavior:**
   - As $$ x \to \infty $$, $$ f(x) $$ behaves like $$ x^{4/3} $$, which increases to infinity.
   - As $$ x \to -\infty $$, $$ f(x) $$ behaves like $$ x^{4/3} $$ but remains positive (since even roots yield positive values).

4. **Additional Points:**
   - Plot a few more points for accuracy, such as $$ x = 4 $$ (where $$ f(4) = 4^{1/3}(0) = 0 $$).

**Note:** The actual graph may require more detailed plotting for precision, especially around $$ x = 0 $$ due to the cube root function.
The critical points of $$ f(x) = x^{1/3}(x - 4) $$ are at $$ x = 0 $$ and $$ x = 1 $$. The graph of $$ f(x) $$ has a critical point at $$ (0, 0) $$ and another at $$ (1, -3) $$.

(c) Find the critical points of \( f(x)=x^{1 / 3}(x-4) \). Sketch the graph of \( f(x) \).

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