The function given is \( y = x^3 - 3x^2 \).

Let's analyze this function.

### 1. **Finding the first derivative:**
The first derivative of the function \( y \) with respect to \( x \) will give us the slope of the tangent to the curve at any point \( x \).

\[
\frac{dy}{dx} = 3x^2 - 6x
\]

### 2. **Finding the critical points:**
Critical points occur where the first derivative is zero or undefined. Setting the first derivative equal to zero:

\[
3x^2 - 6x = 0
\]

Factor the equation:

\[
3x(x - 2) = 0
\]

So, the critical points are \( x = 0 \) and \( x = 2 \).

### 3. **Second derivative to determine concavity:**
The second derivative will help us determine the concavity of the function and whether the critical points are maxima, minima, or points of inflection.

\[
\frac{d^2y}{dx^2} = 6x - 6
\]

Evaluate the second derivative at the critical points:

- At \( x = 0 \):

\[
\frac{d^2y}{dx^2} = 6(0) - 6 = -6 \quad (\text{concave down, indicating a local maximum})
\]

- At \( x = 2 \):

\[
\frac{d^2y}{dx^2} = 6(2) - 6 = 6 \quad (\text{concave up, indicating a local minimum})
\]

### 4. **Finding the points:**
Now, let's find the corresponding \( y \) values for the critical points:

- At \( x = 0 \):

\[
y = 0^3 - 3(0)^2 = 0
\]

So, \( (0, 0) \) is a local maximum.

- At \( x = 2 \):

\[
y = 2^3 - 3(2)^2 = 8 - 12 = -4
\]

So, \( (2, -4) \) is a local minimum.

### 5. **Summary:**
- The function has a local maximum at \( (0, 0) \).
- The function has a local minimum at \( (2, -4) \).

Would you like more details on any of these steps or have any questions?

Here are 5 related questions:
1. How do you find inflection points for the function \( y = x^3 - 3x^2 \)?
2. What is the behavior of the function as \( x \) approaches positive and negative infinity?
3. How do you determine the concavity of the function across different intervals?
4. Can you graph the function \( y = x^3 - 3x^2 \) and indicate the critical points?
5. How do you solve optimization problems using derivatives?

**Tip:** Always check the second derivative when analyzing critical points to confirm whether they are maxima, minima, or inflection points.

Explore the function y = x^3 - 3x^2 with detailed steps on finding derivatives, critical points, and determining local extrema. This problem covers calculus topics including derivatives, concavity, and local extrema analysis. Suitable for advanced high school students and calculus enthusiasts.

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