https://file.aiquickdraw.com/mathbot/file/172724105545166sgcu5t.jpg

The figure shows the graph of f. 
Is f continuous at x = 1?  (Enter Y or N.) 
Is f continuous at x = 2?   
Is f continuous at x = 3?   
Is f continuous at x = 4?  
Is f differentiable at x = 1?  
Is f differentiable at x = 2?  
Is f differentiable at x = 3?  
Is f differentiable at x = 4?  
 
The figure shows the graph of g. 
Is g differentiable at x = 1?  
Is g differentiable at x = 2?  
Is g differentiable at x = 3?  
Is g differentiable at x = 4? 

Let's analyze the graphs of \( f(x) \) and \( g(x) \) to answer the continuity and differentiability questions.

### Continuity of \( f(x) \):

- **At \( x = 1 \)**: The graph shows a jump discontinuity at \( x = 1 \), where the function jumps from an open circle to a filled circle. Therefore, **\( f(x) \) is not continuous at \( x = 1 \)**.
- **At \( x = 2 \)**: The graph is continuous at \( x = 2 \), as there are no breaks or jumps. **\( f(x) \) is continuous at \( x = 2 \)**.
- **At \( x = 3 \)**: Similarly, there is no break in the graph at \( x = 3 \), so **\( f(x) \) is continuous at \( x = 3 \)**.
- **At \( x = 4 \)**: The graph is continuous at \( x = 4 \), with no jump or discontinuity. **\( f(x) \) is continuous at \( x = 4 \)**.

### Differentiability of \( f(x) \):

- **At \( x = 1 \)**: Since \( f(x) \) is not continuous at \( x = 1 \), it cannot be differentiable there. **\( f(x) \) is not differentiable at \( x = 1 \)**.
- **At \( x = 2 \)**: The graph has a sharp corner at \( x = 2 \), which means the derivative is not defined. **\( f(x) \) is not differentiable at \( x = 2 \)**.
- **At \( x = 3 \)**: The graph is smooth at \( x = 3 \), so it is differentiable. **\( f(x) \) is differentiable at \( x = 3 \)**.
- **At \( x = 4 \)**: There is a sharp corner at \( x = 4 \), so **\( f(x) \) is not differentiable at \( x = 4 \)**.

### Differentiability of \( g(x) \):

- **At \( x = 1 \)**: The graph is smooth at \( x = 1 \), so **\( g(x) \) is differentiable at \( x = 1 \)**.
- **At \( x = 2 \)**: There is a sharp corner at \( x = 2 \), so **\( g(x) \) is not differentiable at \( x = 2 \)**.
- **At \( x = 3 \)**: The graph is smooth at \( x = 3 \), so **\( g(x) \) is differentiable at \( x = 3 \)**.
- **At \( x = 4 \)**: Again, a sharp corner appears at \( x = 4 \), so **\( g(x) \) is not differentiable at \( x = 4 \)**.

### Summary of Answers:
- \( f \) continuous at \( x = 1 \): N
- \( f \) continuous at \( x = 2 \): Y
- \( f \) continuous at \( x = 3 \): Y
- \( f \) continuous at \( x = 4 \): Y
- \( f \) differentiable at \( x = 1 \): N
- \( f \) differentiable at \( x = 2 \): N
- \( f \) differentiable at \( x = 3 \): Y
- \( f \) differentiable at \( x = 4 \): N
- \( g \) differentiable at \( x = 1 \): Y
- \( g \) differentiable at \( x = 2 \): N
- \( g \) differentiable at \( x = 3 \): Y
- \( g \) differentiable at \( x = 4 \): N

Let me know if you'd like further explanation on any specific concept!

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Here are 5 relative questions:
1. What is the relationship between continuity and differentiability?
2. How do sharp corners affect the differentiability of a function?
3. Can a function be differentiable but not continuous? Why or why not?
4. How do we interpret open and closed circles on a graph in terms of function values?
5. What is the importance of the derivative in analyzing the smoothness of a curve?

**Tip:** If a function has a sharp corner or cusp, it means the left-hand and right-hand derivatives differ, making it non-differentiable at that point.

The figure shows the graph of f. Is f continuous at x = 1? Is f continuous at x = 2? Is f continuous at x = 3? Is f continuous at x = 4? Is f differentiable at x = 1? Is f differentiable at x = 2? Is f differentiable at x = 3? Is f differentiable at x = 4? The figure shows the graph of g. Is g differentiable at x = 1? Is g differentiable at x = 2? Is g differentiable at x = 3? Is g differentiable at x = 4?

This problem asks whether the function f(x) is continuous and differentiable at points x = 1, 2, 3, and 4 based on the graph of the function. It also asks about the differentiability of g(x) at specific points, focusing on the concept of sharp corners and continuity.

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