If `f(x)=|x-1|+|x+4|x-9|+….+|x-2500| AA x in R`, then all the values of x where f(x) has minimum values lie in

Options

A. (600, 700)
B. (576, 678)
C. (625, 678)
D. none of these

Explanation

To solve the problem, we need to analyze the function \( f(x) = |x - 1| + |x + 4| + |x - 9| + \ldots + |x - 2500| \) and determine where it achieves its minimum values.

### Step-by-Step Solution:

1. **Identify the Points of Non-Differentiability**:
The function \( f(x) \) consists of several absolute value terms. Each term \( |x - a| \) is not differentiable at \( x = a \). Therefore, we need to identify the points where \( f(x) \) is not differentiable:
- The critical points are \( x = 1, -4, 9, \ldots, 2500 \).

2. **List the Critical Points**:
The critical points can be listed as follows:
- \( 1, -4, 9, 16, 25, \ldots, 2500 \)
- These points can be expressed as \( n^2 \) for \( n = 1, 2, \ldots, 50 \) (since \( 50^2 = 2500 \)).

3. **Determine the Range of Minimum Values**:
The minimum value of \( f(x) \) occurs between the smallest and largest critical points. The smallest critical point is \( 1^2 = 1 \) and the largest is \( 50^2 = 2500 \).

4. **Find the Midpoint**:
To find where \( f(x) \) achieves its minimum, we look at the midpoint of the range defined by the critical points:
- The midpoint between \( 1 \) and \( 2500 \) is \( \frac{1 + 2500}{2} = 1250.5 \).

5. **Conclusion**:
Since the function is symmetric around the midpoint, the minimum values of \( f(x) \) will occur in the interval between the critical points \( 1 \) and \( 2500 \). Thus, the values of \( x \) where \( f(x) \) has minimum values lie in the interval \( [1, 2500] \).

### Final Answer:
The values of \( x \) where \( f(x) \) has minimum values lie in the interval \( [1, 2500] \).

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