Explanation
The correct answer is (d) Neither I nor II.
1. Definition of Extrema on Open Intervals:
A function f(x) attains a maximum (or minimum) value in an interval if there exists some value c within that interval such that f(c)≥f(x) (or f(c)≤f(x)) for all x in that interval.
2. Analyzing the Function f(x)=x:
The function f(x)=x is a strictly increasing linear function. In the open interval (−1,1), the values of x get arbitrarily close to −1 from the right and arbitrarily close to 1 from the left, but the function never actually reaches −1 or 1 because these endpoints are excluded from the interval.
3. Why both statements are false:
No Maximum: For any value c∈(−1,1), you can always choose a value x closer to 1 such that x > c. Therefore, f(x) > f(c), and no absolute maximum exists.
No Minimum: Similarly, for any value c∈(−1,1), you can always choose a value x closer to −1 such that x < c. Therefore, f(x) < f(c), and no absolute minimum exists.
Conclusion:
Since the interval (−1,1) is an open interval, the function f(x)=x does not attain a maximum or a minimum value within this domain.
