Minimum?
Find the minimum value of the following expression:
Notation : denotes the absolute value function .
The answer is 6.
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1 solution
The question is equivalent to finding a point x on number line such that the sum of distance between x and 1, 2, 3, 4, 5 is minimized.
Notice that the sum of distance between x and 1, and between x and 5 is minimized when x is between 1 and 5. In that case, the sum of distance between x and 1, and between x and 5 is equal to 5 − 1 = 4 .
Also notice that the sum of distance between x and 2, and between x and 4 is minimized when x is between 2 and 4. In that case, the sum of distance between x and 2, and between x and 4 is equal to 4 − 2 = 2 .
Besides, notice that the distance between x and 3 is minimized when x = 3 .
Since the location of x which minimized the sum of distance in the above three cases intersect at x = 3 , x = 3 minimize the given expression.
So, the minimum value is 2 + 1 + 0 + 1 + 2 = 6 .