The minimum possible value of - x -1- x -2- x -100- isA- 4900B- 2500- 2450D- 2550

The correct option is B 2500Critical points are 1,2,3,⋯,99,100 Minimum value occurs at x=median{1,2,3,⋯,99,100}=50 or 51 For x=50, |x 1|+|x 2|+⋯+|x 100| =49+4 ...

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Standard XII

Mathematics

Absolute Value Function

The minimum p...

Question

The minimum possible value of $| x - 1 | + | x - 2 | + \dots + | x - 100 |$ is

4900

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2500

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2550

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2450

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Solution

The correct option is B $2500$
Critical points are $1, 2, 3, \dots, 99, 100$
Minimum value occurs at $x = median {1, 2, 3, \dots, 99, 100} = 50 or 51$
For $x = 50$ ,
$| x - 1 | + | x - 2 | + \dots + | x - 100 |$
$= 49 + 48 + \dots + 1 + 0 + 1 + \dots + 49 + 50$
$= 2 (1 + 2 + \dots + 49) + 50$
$2500$

Note: The minimum value occurs at any value in $[50, 51]$

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The minimum possible value of |x−1|+|x−2|+⋯+|x−100| is

The correct option is B 2500Critical points are 1,2,3,⋯,99,100 Minimum value occurs at x=median{1,2,3,⋯,99,100}=50 or 51 For x=50, |x−1|+|x−2|+⋯+|x−100| =49+48+⋯+1+0+1+⋯+49+50 =2(1+2+⋯+49)+50 2500 Note: The minimum value occurs at any value in [50,51]

The minimum possible value of $| x - 1 | + | x - 2 | + \dots + | x - 100 |$ is