Function Analysis: \(f(x) = x\)

The function given is \(f(x) = x\). This is a simple linear function that increases as \(x\) increases.

Interval Consideration: Open Interval (-1, 1)

The interval specified is (-1, 1). This is an open interval, meaning it includes all numbers between -1 and 1, but not -1 and 1 themselves. Mathematically, \(x \in (-1, 1)\) means \(-1 < x < 1\).

Statement I: Maximum Value Attainment

In the interval (-1, 1), the values of \(f(x) = x\) range strictly between -1 and 1.

• As \(x\) gets closer and closer to 1 (e.g., 0.9, 0.99, 0.999), \(f(x)\) gets closer to 1.
• However, since \(x\) must be less than 1, \(f(x)\) can never be exactly equal to 1.
• Therefore, the function does not attain a maximum value within the open interval (-1, 1). The supremum is 1, but it is not reached.

Statement II: Minimum Value Attainment

Similarly, in the interval (-1, 1):

• As \(x\) gets closer and closer to -1 (e.g., -0.9, -0.99, -0.999), \(f(x)\) gets closer to -1.
• However, since \(x\) must be greater than -1, \(f(x)\) can never be exactly equal to -1.
• Therefore, the function does not attain a minimum value within the open interval (-1, 1). The infimum is -1, but it is not reached.

Conclusion

Since neither statement I nor statement II is correct, the correct option is that the function attains neither its maximum nor its minimum value in the given open interval.

Correct Answer: Neither I nor II