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The problem asks us to determine whether the function \( f(x) \) is continuous and differentiable at \( x = 2 \). The function is defined as a piecewise function:

\[
f(x) = 
\begin{cases} 
x^2 & \text{if } x \leq 2 \\
12 - x & \text{if } x > 2 
\end{cases}
\]

### A. Check if \( f(x) \) is continuous at \( x = 2 \):

A function is continuous at \( x = 2 \) if:

1. \( f(2) \) is defined.
2. The left-hand limit \( \lim_{x \to 2^-} f(x) \) exists.
3. The right-hand limit \( \lim_{x \to 2^+} f(x) \) exists.
4. Both limits are equal to \( f(2) \).

#### 1. Evaluate \( f(2) \):

Since \( 2 \leq 2 \), use the first piece of the function:

\[
f(2) = 2^2 = 4.
\]

#### 2. Evaluate the left-hand limit \( \lim_{x \to 2^-} f(x) \):

For \( x \leq 2 \), use \( f(x) = x^2 \):

\[
\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x^2 = 2^2 = 4.
\]

#### 3. Evaluate the right-hand limit \( \lim_{x \to 2^+} f(x) \):

For \( x > 2 \), use \( f(x) = 12 - x \):

\[
\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (12 - x) = 12 - 2 = 10.
\]

#### 4. Compare limits and \( f(2) \):

\[
\lim_{x \to 2^-} f(x) = 4, \quad \lim_{x \to 2^+} f(x) = 10, \quad f(2) = 4.
\]

Since the left-hand limit is not equal to the right-hand limit, the function is **not continuous** at \( x = 2 \).

### B. Check if \( f(x) \) is differentiable at \( x = 2 \):

A function is differentiable at \( x = 2 \) only if it is continuous there. Since \( f(x) \) is not continuous at \( x = 2 \), it is also **not differentiable** at \( x = 2 \).

### Conclusion:
- **A.** \( f(x) \) is **not continuous** at \( x = 2 \).
- **B.** \( f(x) \) is **not differentiable** at \( x = 2 \).

Would you like a more detailed explanation or have any questions?

Here are some related questions that might help you understand this topic better:

1. How do you determine if a function is continuous at a point?
2. What are the necessary conditions for a function to be differentiable at a point?
3. How do you find the left-hand and right-hand limits of a piecewise function?
4. What is the difference between continuity and differentiability?
5. Can a function be differentiable if it is not continuous?

**Tip:** A function must be continuous at a point to be differentiable there, but continuity does not necessarily imply differentiability.

Determine whether the function f(x) is continuous and differentiable at x = 2. The function is defined as f(x) = {x^2 for x <= 2, 12 - x for x > 2}.

Determine whether the function f(x), defined as a piecewise function, is continuous and differentiable at x = 2. Learn about continuity and differentiability with a detailed explanation.

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