To find the critical numbers of the function $f(x) = \sqrt[3]{2x^2 - 18}$, we need to follow these steps:

1. Find the first derivative of $f(x)$.
2. Set the first derivative equal to zero and solve for $x$ to find critical points.
3. Check where the first derivative is undefined.
4. Determine the local minimum and maximum points using a graph.

Step 1: Find the First Derivative

The function is $f(x) = (2x^2 - 18)^{1/3}$.

Using the chain rule for differentiation: $f'(x) = \frac{d}{dx} \left( (2x^2 - 18)^{1/3} \right)$

Let $u = 2x^2 - 18$. Then $f(x) = u^{1/3}$.

Using the chain rule: $f'(x) = \frac{d}{du} \left( u^{1/3} \right) \cdot \frac{du}{dx}$ $f'(x) = \frac{1}{3} u^{-2/3} \cdot (4x)$ $f'(x) = \frac{4x}{3} (2x^2 - 18)^{-2/3}$

Step 2: Set the First Derivative Equal to Zero and Solve for $x$

$\frac{4x}{3} (2x^2 - 18)^{-2/3} = 0$

This equation is zero when $4x = 0$: $x = 0$

Step 3: Check Where the First Derivative is Undefined

The first derivative is undefined when the denominator is zero: $(2x^2 - 18)^{-2/3} = \infty$ This occurs when: $2x^2 - 18 = 0$ $x^2 = 9$ $x = \pm 3$

Critical Numbers

The critical numbers are $x = 0, 3,$ and $-3$.

Step 4: Determine Local Minimum and Maximum Points Using a Graph

We need to analyze the behavior of $f(x)$ at these critical points using the first and second derivative tests, but we can also use a graph to visually determine the local minimum and maximum points.

Plotting $f(x) = \sqrt[3]{2x^2 - 18}$:

By graphing, we can observe the behavior of the function near the critical points.

Critical Point Analysis

• At $x = 0$:

  • $f(0) = \sqrt[3]{2(0)^2 - 18} = \sqrt[3]{-18}$
  • The function has an inflection point here since the graph changes concavity but does not achieve a local minimum or maximum.

• At $x = 3$:

  • $f(3) = \sqrt[3]{2(3)^2 - 18} = \sqrt[3]{0} = 0$
  • The function achieves a local minimum here.

• At $x = -3$:

  • $f(-3) = \sqrt[3]{2(-3)^2 - 18} = \sqrt[3]{0} = 0$
  • The function achieves a local maximum here.

Conclusion

• Critical numbers: $x = 0, 3, -3$
• Local minimum point: $(3, 0)$
• Local maximum point: $(-3, 0)$

Would you like further details or have any questions?

Here are five questions you might ask:

1. How do you perform the first and second derivative tests?
2. Can you explain how to graph a function using critical points?
3. What is the significance of an inflection point?
4. How do you determine if a function is increasing or decreasing?
5. What are some common mistakes to avoid when finding critical points?

Tip: Always check where the derivative is undefined as these points can be critical for identifying the behavior of the function.