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The function \( f(x) \) is defined piecewise as:

\[
f(x) = \begin{cases} 
x^2 - 20 & \text{for } x < 5 \\
-x^2 + 20 & \text{for } x \geq 5
\end{cases}
\]

### Step 1: Check Continuity at \( x = 5 \)

For \( f(x) \) to be differentiable at \( x = 5 \), it must first be continuous at \( x = 5 \). Let's check if the left-hand limit (\( x \to 5^- \)) and the right-hand limit (\( x \to 5^+ \)) agree, and if they match the value of \( f(5) \).

- **Left-hand limit** as \( x \to 5^- \):
  \[
  \lim_{x \to 5^-} f(x) = \lim_{x \to 5^-} (x^2 - 20) = 5^2 - 20 = 25 - 20 = 5
  \]
  
- **Right-hand limit** as \( x \to 5^+ \):
  \[
  \lim_{x \to 5^+} f(x) = \lim_{x \to 5^+} (-x^2 + 20) = -(5^2) + 20 = -25 + 20 = -5
  \]

Since the left-hand limit is \( 5 \) and the right-hand limit is \( -5 \), the limits do not match. Therefore, \( f(x) \) is not continuous at \( x = 5 \).

### Conclusion:
Since the function is not continuous at \( x = 5 \), it cannot be differentiable there. Therefore, **option A** is correct: 
- \( f \) is not differentiable at \( x = 5 \) because \( f \) is not continuous at \( x = 5 \).

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Would you like a detailed explanation of any particular part? Let me know! Here are 5 questions related to this topic:

1. What conditions must a function satisfy to be differentiable at a point?
2. How can you determine the continuity of a piecewise function at a specific point?
3. Why is continuity a necessary condition for differentiability?
4. What is the significance of one-sided limits in determining continuity at a point?
5. How does the presence of a sharp corner or cusp affect differentiability?

**Tip:** Always check the continuity of a function at a point before checking for differentiability!

Let f be the function defined above. Which of the following statements is true?

A) f is not differentiable at x = 5 because f is not continuous at x = 5.
B) f is not differentiable at x = 5 because the graph of f has a sharp corner at x = 5.
C) f is not differentiable at x = 5 because the graph of f has a vertical tangent at x = 5.
D) f is not differentiable at x = 5 because f is not defined at x = 5.

Explore the differentiability and continuity of a piecewise function at x = 5. This problem involves understanding the left-hand and right-hand limits, and applying the conditions for continuity and differentiability. Suitable for students in Grades 11-12.

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