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Let's work through the problem step by step.

We are given the piecewise function:

\[
f(x) = 
\begin{cases} 
x^2 - 4x & \text{if } x \leq 6 \\
2x & \text{if } x > 6 
\end{cases}
\]

### (a) Is \( f \) continuous at \( x = 6 \)?

To check continuity at \( x = 6 \), we need to verify if:

\[
\lim_{x \to 6^-} f(x) = \lim_{x \to 6^+} f(x) = f(6)
\]

1. **Left-hand limit** \( \lim_{x \to 6^-} f(x) \):  
For \( x \leq 6 \), \( f(x) = x^2 - 4x \).  
So,

\[
\lim_{x \to 6^-} f(x) = 6^2 - 4(6) = 36 - 24 = 12
\]

2. **Right-hand limit** \( \lim_{x \to 6^+} f(x) \):  
For \( x > 6 \), \( f(x) = 2x \).  
So,

\[
\lim_{x \to 6^+} f(x) = 2(6) = 12
\]

3. **Value at \( x = 6 \)**:  
We use the piece \( f(x) = x^2 - 4x \) since \( x = 6 \) is included in the first part of the piecewise function.  
So,

\[
f(6) = 6^2 - 4(6) = 12
\]

Since the left-hand limit, right-hand limit, and \( f(6) \) are all equal to 12, **\( f \) is continuous at \( x = 6 \)**.

### (b) Is \( f \) differentiable at \( x = 6 \)?

To check differentiability, we need to verify if the left-hand derivative and right-hand derivative at \( x = 6 \) are equal.

1. **Left-hand derivative** (for \( x \leq 6 \)):  
For \( f(x) = x^2 - 4x \), the derivative is:

\[
f'(x) = 2x - 4
\]

At \( x = 6 \),

\[
f'(6^-) = 2(6) - 4 = 12 - 4 = 8
\]

2. **Right-hand derivative** (for \( x > 6 \)):  
For \( f(x) = 2x \), the derivative is:

\[
f'(x) = 2
\]

At \( x = 6 \),

\[
f'(6^+) = 2
\]

Since the left-hand derivative \( f'(6^-) = 8 \) and the right-hand derivative \( f'(6^+) = 2 \) are not equal, **\( f \) is not differentiable at \( x = 6 \)**.

### (c) Since \( f \) is not differentiable at \( x = 6 \), we enter the value of \( f(6) \):

From part (a), we found that:

\[
f(6) = 12
\]

So the answer is \( f(6) = 12 \).

---

Would you like further details or have any questions on this?

Here are some related questions:

1. How do you define the conditions for continuity in a piecewise function?
2. What does it mean for a function to be differentiable at a point?
3. Can a function be continuous but not differentiable at a point? Why?
4. What are some common points of non-differentiability in piecewise functions?
5. How can we interpret the graphical behavior of a function that is continuous but not differentiable?

**Tip:** When working with piecewise functions, always check continuity first before checking differentiability at the points where the function changes its rule.

Is the function f(x), defined piecewise with f(x) = x^2 - 4x for x <= 6 and f(x) = 2x for x > 6, continuous and differentiable at x = 6?

This problem explores whether a piecewise function, f(x) = x^2 - 4x for x <= 6 and f(x) = 2x for x > 6, is continuous and differentiable at x = 6. Includes analysis using limits and derivatives.

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