Given the piecewise function \( f(x) \), let's analyze it:

\[
f(x) = \begin{cases} 
x^3 + 5 & \text{if } x < 1, \\
7x - x^3 & \text{if } x \geq 1.
\end{cases}
\]

### 1. **Continuity at \( x = 1 \):**
To check whether \( f(x) \) is continuous at \( x = 1 \), we need to verify that the left-hand limit, the right-hand limit, and the value of the function at \( x = 1 \) are all equal.

- **Left-hand limit** as \( x \) approaches 1:
  \[
  \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x^3 + 5) = 1^3 + 5 = 6
  \]
  
- **Right-hand limit** as \( x \) approaches 1:
  \[
  \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (7x - x^3) = 7(1) - 1^3 = 7 - 1 = 6
  \]
  
- **Value of the function** at \( x = 1 \):
  \[
  f(1) = 7(1) - 1^3 = 7 - 1 = 6
  \]

Since the left-hand limit, the right-hand limit, and the value of the function at \( x = 1 \) are all equal, \( f(x) \) is continuous at \( x = 1 \).

### 2. **Differentiability at \( x = 1 \):**
Now let's check whether the derivative exists at \( x = 1 \).

- **Left-hand derivative** as \( x \) approaches 1:
  \[
  f'(x) = \frac{d}{dx}(x^3 + 5) = 3x^2
  \]
  \[
  \lim_{x \to 1^-} f'(x) = 3(1)^2 = 3
  \]

- **Right-hand derivative** as \( x \) approaches 1:
  \[
  f'(x) = \frac{d}{dx}(7x - x^3) = 7 - 3x^2
  \]
  \[
  \lim_{x \to 1^+} f'(x) = 7 - 3(1)^2 = 7 - 3 = 4
  \]

Since the left-hand derivative \( 3 \) and the right-hand derivative \( 4 \) are not equal, the derivative does not exist at \( x = 1 \).

### Conclusion:
- \( f(x) \) is continuous at \( x = 1 \), but it is not differentiable at \( x = 1 \).

Would you like more details or have any further questions?

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Here are 5 related questions:

1. How do you determine whether a piecewise function is differentiable at a point where the definition changes?
2. Can you give another example of a piecewise function that is continuous but not differentiable at a certain point?
3. How can we graphically interpret the discontinuity in the derivative at a point?
4. What is the geometric interpretation of a function's continuity at a point?
5. Can a function be differentiable at a point if it is not continuous at that point?

**Tip:** When dealing with piecewise functions, always check for both continuity and differentiability at the points where the function definition changes.

Explore the analysis of the piecewise function f(x) involving limits and derivatives at the point x = 1. This mathematical problem covers continuity, differentiability, and key theorems like the Intermediate Value Theorem. Suitable for advanced high school students.

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