applications of derivatives

Question

The minimum value of $f(x) = |x - 1| + |x - 2| + ... + |x - 2025|$ occurs at:

RankGuru · Accepted Answer

For a function of the form $f(x) = \sum |x - a_i|$ where $a_i$ are sorted increasing order, the minimum occurs at the median of the sequence $a_i$.

Here, the sequence is $1, 2, ..., 2025$.
The total number of terms is 2025, which is an odd number.
The median is the $\frac{n+1}{2}$-th term.
Median position $= \frac{2025 + 1}{2} = 1013$.

Thus, the minimum occurs at $x = 1013$.

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