The problem asks for determining the local extremum points of the function $f(x) = x^2 - x^3$, along with identifying the nature of these points.

Step 1: Find the derivative of the function

To find the local extremum points, we first need to find the first derivative of the function $f(x)$:

$f(x) = x^2 - x^3$

Differentiate with respect to $x$:

$f'(x) = 2x - 3x^2$

Step 2: Find the critical points

To find the critical points, set the first derivative equal to zero and solve for $x$:

$f'(x) = 2x - 3x^2 = 0$

Factor the equation:

$x(2 - 3x) = 0$

This gives two solutions:

$x = 0 \quad \text{or} \quad x = \frac{2}{3}$

So, the critical points are $x = 0$ and $x = \frac{2}{3}$.

Step 3: Determine the nature of the critical points

To determine whether these points are local minima, maxima, or saddle points, we examine the second derivative of the function $f(x)$.

First, find the second derivative of $f(x)$:

$f''(x) = 2 - 6x$

• At $x = 0$:

$f''(0) = 2 - 6(0) = 2$

Since $f''(0) > 0$, the function has a local minimum at $x = 0$.

• At $x = \frac{2}{3}$:

$f''\left(\frac{2}{3}\right) = 2 - 6\left(\frac{2}{3}\right) = 2 - 4 = -2$

Since $f''\left(\frac{2}{3}\right) < 0$, the function has a local maximum at $x = \frac{2}{3}$.

Conclusion

• There is a local minimum at $x = 0$.
• There is a local maximum at $x = \frac{2}{3}$.

Would you like more details or explanations on any part of the solution? Here are some related questions you might be interested in:

1. What is the graphical interpretation of local extrema?
2. How do we classify points if the second derivative test is inconclusive?
3. Can you find the global maximum or minimum of this function?
4. How does the function behave at the boundaries (as $x$ approaches infinity)?
5. What is the significance of inflection points in the context of this function?

Tip: When finding extremum points, always double-check your critical points using the second derivative or another appropriate test.