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We are analyzing the function \( f(x) = x^3 - 3x^2 + 2 \), and the task involves determining the following characteristics:

1. **Continuity**: 
   The function \( f(x) = x^3 - 3x^2 + 2 \) is a polynomial, which means it is continuous everywhere in its domain. Therefore, the function is **continuous**.

2. **End Behavior**: 
   To describe the end behavior, we examine the limits of the function as \( x \) approaches positive and negative infinity:
   \[
   \lim_{x \to -\infty} f(x) = -\infty
   \]
   \[
   \lim_{x \to \infty} f(x) = \infty
   \]
   So, as \( x \to -\infty \), \( y \to -\infty \), and as \( x \to \infty \), \( y \to \infty \).

3. **Increasing and Decreasing Intervals**:
   To determine where the function is increasing or decreasing, we take the derivative of the function:
   \[
   f'(x) = 3x^2 - 6x
   \]
   Setting \( f'(x) = 0 \) to find critical points:
   \[
   3x(x - 2) = 0 \implies x = 0 \text{ or } x = 2
   \]
   These critical points divide the domain into intervals. Let's test the sign of \( f'(x) \) in each interval:

   - For \( x < 0 \) (e.g., \( x = -1 \)), \( f'(-1) = 3(-1)^2 - 6(-1) = 3 + 6 = 9 > 0 \), so the function is **increasing** on \( (-\infty, 0) \).
   - For \( 0 < x < 2 \) (e.g., \( x = 1 \)), \( f'(1) = 3(1)^2 - 6(1) = 3 - 6 = -3 < 0 \), so the function is **decreasing** on \( (0, 2) \).
   - For \( x > 2 \) (e.g., \( x = 3 \)), \( f'(3) = 3(3)^2 - 6(3) = 27 - 18 = 9 > 0 \), so the function is **increasing** on \( (2, \infty) \).

4. **Constant Intervals**: 
   There are no intervals where the function is constant.

### Summary:
- **Continuous**: Yes, the function is continuous.
- **End Behavior**: 
   - As \( x \to -\infty \), \( y \to -\infty \).
   - As \( x \to \infty \), \( y \to \infty \).
- **Increasing**: \( (-\infty, 0) \cup (2, \infty) \).
- **Decreasing**: \( (0, 2) \).
- **Constant**: None.

Would you like more details, or have any questions?

Here are five related questions for further understanding:
1. How do you find the derivative of a function to determine critical points?
2. What is the significance of the critical points in determining increasing and decreasing intervals?
3. Can you explain why polynomial functions are always continuous?
4. How does concavity relate to increasing and decreasing behavior of functions?
5. What methods can you use to analyze more complex functions?

**Tip**: Always graph the function or use its derivative to visualize where it is increasing, decreasing, or has turning points.

Determine if the function f(x) = x^3 - 3x^2 + 2 is continuous or discontinuous, describe the end behavior, and identify where the function is increasing, decreasing, or constant.

Analyze the polynomial function f(x) = x^3 - 3x^2 + 2 to determine its continuity, end behavior, and intervals of increase and decrease. The solution explores key calculus concepts like the derivative, critical points, and end behavior at infinity.

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