Correct Answer: (C)

1. Conceptual Overview: The function f(x) is the sum of absolute differences between a point x and a set of fixed points (1, 2, 3, ..., 100). In statistics and coordinate geometry, the sum of absolute deviations from a point x is minimized when x is the median of the distribution.

2. Identifying the Median: The set of points consists of 100 terms, which is an even number. For an even-numbered set of data points arranged in increasing order (a_1, a_2, ..., a_n), the median is not a single point but any value in the closed interval between the two central terms.

3. Determining the Central Terms: For n = 100, the two middle terms are the (n/2)th term and the (n/2 + 1)th term. Calculating these gives the 50th term (50) and the 51st term (51).

4. Mathematical Verification: Consider the derivative of f(x). For any term |x - a|, the derivative is -1 if x < a and +1 if x > a. For f(x) to be at a minimum, the number of positive slopes must equal the number of negative slopes.

5. Slope Analysis: When x is between 50 and 51, there are exactly 50 terms to the left (1 through 50) providing a slope of +1 each, and 50 terms to the right (51 through 100) providing a slope of -1 each. The net slope is 50 - 50 = 0.

6. Conclusion: Since the derivative is zero for all x such that 50 < x < 51, and the function is continuous, the minimum value is constant across the entire interval [50, 51].

Test Prep Tip: In problems involving the sum of absolute values, always look for the median. If the number of points is odd, the minimum is at the middle point; if even, the minimum is any value in the range between the two middle points.