https://file.aiquickdraw.com/mathbot/file/1726978375068virdbhsr.jpg

The image describes a piecewise function \( f(x) \) defined as:

\[
f(x) = 
\begin{cases} 
x^2 - 4, & x \leq 2 \\
k(x - 2), & x > 2
\end{cases}
\]

The questions are:

### (a) For what values of \( k \) is \( f \) continuous?

For continuity at \( x = 2 \), the left-hand limit and the right-hand limit must be equal to the value of the function at \( x = 2 \). Let's evaluate this:

1. The left-hand limit as \( x \to 2^- \) is:
   \[
   \lim_{x \to 2^-} f(x) = 2^2 - 4 = 0
   \]
   
2. The right-hand limit as \( x \to 2^+ \) is:
   \[
   \lim_{x \to 2^+} f(x) = k(2 - 2) = 0
   \]
   
3. The value of the function at \( x = 2 \) is \( f(2) = 2^2 - 4 = 0 \).

Since all limits and the function value at \( x = 2 \) are \( 0 \), the function is continuous for **all values of \( k \)**.

### (b) For what values of \( k \) is \( f \) differentiable?

For differentiability at \( x = 2 \), the left-hand and right-hand derivatives must be equal. Let's compute both:

1. The derivative of \( f(x) = x^2 - 4 \) for \( x \leq 2 \) is:
   \[
   f'(x) = 2x
   \]
   So the left-hand derivative as \( x \to 2^- \) is:
   \[
   \lim_{x \to 2^-} f'(x) = 2(2) = 4
   \]
   
2. The derivative of \( f(x) = k(x - 2) \) for \( x > 2 \) is:
   \[
   f'(x) = k
   \]
   So the right-hand derivative as \( x \to 2^+ \) is:
   \[
   \lim_{x \to 2^+} f'(x) = k
   \]

For \( f(x) \) to be differentiable at \( x = 2 \), the left-hand and right-hand derivatives must be equal. Therefore:
\[
4 = k
\]

Thus, \( f(x) \) is differentiable when \( k = 4 \).

---

Let me know if you'd like further explanation on any part of this solution!

### 5 Related Questions:
1. How do you check the continuity of a piecewise function at the boundary point?
2. What is the importance of left-hand and right-hand derivatives for differentiability?
3. How does the differentiability of a function imply continuity?
4. Can a function be continuous but not differentiable? If so, when?
5. What happens to the continuity or differentiability when \( k \neq 4 \)?

### Tip:
When solving for continuity and differentiability, always compute limits from both sides of the boundary point carefully, ensuring the function behavior is smooth (continuous) and the slope matches (differentiable).

Suppose that f(x) is a piecewise function:
- f(x) = x^2 - 4 for x ≤ 2
- f(x) = k(x - 2) for x > 2
(a) For what values of k is f continuous?
(b) For what values of k is f differentiable?

Explore the conditions for continuity and differentiability of the piecewise function f(x) where f(x) = x^2 - 4 for x ≤ 2 and f(x) = k(x - 2) for x > 2. Understand how to solve for the values of k that make the function continuous and differentiable.

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