What are the maximum and minimum values, if any, of `f(x)=x, x in (0, 1)`?

Answer

To find the maximum and minimum values of the function \( f(x) = x \) for \( x \) in the interval \( (0, 1) \), we can follow these steps:

### Step 1: Identify the function and the interval
The function we are analyzing is \( f(x) = x \), and we are interested in the interval \( (0, 1) \).

### Step 2: Analyze the function
The function \( f(x) = x \) is a linear function, which means it is continuous and has a constant slope. As \( x \) increases from 0 to 1, \( f(x) \) also increases.

### Step 3: Evaluate the endpoints of the interval
Since the interval is open, we need to evaluate the behavior of the function as it approaches the endpoints:
- As \( x \) approaches 0 from the right (i.e., \( x \to 0^+ \)), \( f(x) \) approaches \( f(0) = 0 \).
- As \( x \) approaches 1 from the left (i.e., \( x \to 1^- \)), \( f(x) \) approaches \( f(1) = 1 \).

### Step 4: Determine maximum and minimum values
From our evaluation:
- The minimum value of \( f(x) \) in the interval \( (0, 1) \) is \( 0 \) (as \( x \) approaches 0).
- The maximum value of \( f(x) \) in the interval \( (0, 1) \) is \( 1 \) (as \( x \) approaches 1).

### Conclusion
Thus, the minimum value of \( f(x) \) is \( 0 \) and the maximum value is \( 1 \).