To find the critical points of the function $f(x) = (x^2 (x+1))^{1/3}$, we first need to compute the derivative $f'(x)$ and then determine where this derivative is equal to zero or undefined.

Step 1: Differentiate $f(x)$

Using the chain rule and product rule:

1. Rewrite $f(x)$ as $f(x) = (g(x))^{1/3}$ where $g(x) = x^2 (x+1)$.
2. The derivative using the chain rule is: $f'(x) = \frac{1}{3} (g(x))^{-2/3} g'(x)$

Next, we need to compute $g'(x)$.

Step 2: Differentiate $g(x)$

Using the product rule: $g(x) = x^2 (x + 1)$ $g'(x) = 2x(x + 1) + x^2(1) = 2x^2 + 2x = 2x(x + 1)$

Step 3: Substitute $g'(x)$ into $f'(x)$

Now substitute $g'(x)$: $f'(x) = \frac{1}{3} (x^2 (x + 1))^{-2/3} (2x(x + 1))$

Step 4: Set $f'(x) = 0$

To find critical points: $\frac{1}{3} (x^2 (x + 1))^{-2/3} (2x(x + 1)) = 0$

The term $(x^2 (x + 1))^{-2/3}$ is undefined when $x^2 (x + 1) = 0$.

Step 5: Solve $2x(x + 1) = 0$

1. $2x = 0$ gives $x = 0$.
2. $x + 1 = 0$ gives $x = -1$.

Step 6: Check where $f'(x)$ is undefined

Set $x^2 (x + 1) = 0$:

1. $x^2 = 0$ gives $x = 0$.
2. $x + 1 = 0$ gives $x = -1$.

Both points $x = 0$ and $x = -1$ need to be checked.

Summary of Critical Points

The critical points of the function $f(x) = (x^2 (x+1))^{1/3}$ are:

• $x = 0$
• $x = -1$

Conclusion

The critical points are $x = 0$ and $x = -1$.

Would you like more details on any part of this process, or do you have any questions? Here are some related questions you might consider:

1. How do you determine if the critical points are local minima or maxima?
2. What is the second derivative test for concavity?
3. How would you sketch the graph of $f(x)$?
4. Can you find the values of $f(x)$ at the critical points?
5. What are the implications of the critical points in terms of the function's behavior?

Tip: Always check the second derivative to classify critical points as minima, maxima, or points of inflection.