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To analyze the piecewise function \( f(x) \) defined as:

\[
f(x) = 
\begin{cases} 
4x + x^2 & \text{for } x \leq 2 \\
x^3 + 4 & \text{for } x > 2 
\end{cases}
\]

we need to determine the correct statements among the given options by examining the continuity, limits, and differentiability of \( f(x) \).

### 1. Continuity at \( x = 2 \)
To determine if \( f(x) \) is continuous at \( x = 2 \), we must check if:
   - \( f(2) \) exists.
   - \( \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) \).
   - \( f(2) = \lim_{x \to 2} f(x) \).

#### Calculating \( f(2) \)
Using \( f(x) = 4x + x^2 \) for \( x \leq 2 \):
\[
f(2) = 4(2) + (2)^2 = 8 + 4 = 12
\]

#### Calculating \( \lim_{x \to 2^-} f(x) \)
\[
\lim_{x \to 2^-} f(x) = \lim_{x \to 2} (4x + x^2) = 4(2) + (2)^2 = 12
\]

#### Calculating \( \lim_{x \to 2^+} f(x) \)
\[
\lim_{x \to 2^+} f(x) = \lim_{x \to 2} (x^3 + 4) = (2)^3 + 4 = 8 + 4 = 12
\]

Since \( f(2) = \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = 12 \), the function \( f(x) \) is continuous at \( x = 2 \).

### 2. Limit as \( x \to 3 \)
The limit as \( x \to 3 \) for \( f(x) \) can be directly evaluated using the second piece of the function (\( f(x) = x^3 + 4 \) for \( x > 2 \)):

\[
\lim_{x \to 3} f(x) = 3^3 + 4 = 27 + 4 = 31
\]

So, the limit \( \lim_{x \to 3} f(x) \) exists.

### 3. Differentiability at \( x = 2 \)
To check if \( f(x) \) is differentiable at \( x = 2 \), we need to ensure that the left-hand derivative and right-hand derivative at \( x = 2 \) are equal.

#### Left-hand derivative at \( x = 2 \)
For \( x \leq 2 \), \( f(x) = 4x + x^2 \).
\[
f'(x) = 4 + 2x
\]
\[
\lim_{x \to 2^-} f'(x) = 4 + 2(2) = 4 + 4 = 8
\]

#### Right-hand derivative at \( x = 2 \)
For \( x > 2 \), \( f(x) = x^3 + 4 \).
\[
f'(x) = 3x^2
\]
\[
\lim_{x \to 2^+} f'(x) = 3(2)^2 = 3 \times 4 = 12
\]

Since the left-hand derivative \( (8) \) and the right-hand derivative \( (12) \) at \( x = 2 \) are not equal, \( f(x) \) is **not differentiable** at \( x = 2 \).

### Conclusion
Based on the above analysis:
- \( f(x) \) is continuous.
- \( \lim_{x \to 3} f(x) \) exists.
- \( f(x) \) is **not differentiable** at \( x = 2 \).

The correct answers are:
- **\( f(x) \) is continuous**
- **\( f(x) \) is not differentiable**

Would you like more details on any specific part, or have any questions?

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Here are five related questions to deepen understanding:
1. How can we check the differentiability of a piecewise function at a boundary point?
2. What is the significance of left-hand and right-hand derivatives in determining differentiability?
3. Why does the continuity of a function not guarantee its differentiability?
4. How can piecewise functions be made differentiable at the boundary points?
5. What are the practical applications of piecewise functions in real-world problems?

**Tip:** For piecewise functions, always check the boundary points for continuity and differentiability, as they often lead to discontinuities or non-differentiable points.

Define a piecewise function f(x) and determine the correct statements among the given options for continuity, limits, and differentiability.

Learn how to analyze a piecewise function for continuity, limits, and differentiability. This example includes step-by-step calculations for evaluating the function at boundary points and determining the left-hand and right-hand derivatives. Suitable for Grades 11-12.

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