To find the minimum value of the function (f(x) = |x - 1| + |x - 2| + \dots + |x - 2018|), we use the property of sums of absolute values: the minimum occurs at the median of the terms inside the absolute values.

Key Observations:

• The terms inside the absolute values are (1, 2, \dots, 2018) (sorted).
• There are 2018 terms (even), so the median lies between the 1009th and 1010th terms ((1009) and (1010)). Any (x) between these values gives the same minimum.

Calculation:

Let (x = 1009) (or (1010)) to compute the sum:
Split the sum into two parts:

1. Left part: (k = 1) to (1009): (\sum{k=1}^{1009} |1009 - k| = \sum{m=0}^{1008} m = \frac{1008 \times 1009}{2}).
2. Right part: (k = 1010) to (2018): (\sum{k=1010}^{2018} |1009 - k| = \sum{m=1}^{1009} m = \frac{1009 \times 1010}{2}).

Adding these:
[ \text{Total sum} = \frac{1008 \times 1009}{2} + \frac{1009 \times 1010}{2} = \frac{1009}{2}(1008 + 1010) = 1009 \times 1009 = 1009^2. ]

Result:

(1009^2 = 1018081).

Answer: (\boxed{1018081})

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