Minimum value of the expression $| x - 1 | + | x - 2 | + | x - 3 |$ is

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Solution

The correct option is C $2$
Case $1 :$ When $x < 1$
$| x - 1 | + | x - 2 | + | x - 3 |$ becomes
$\Rightarrow f (x) = - x + 1 - x + 2 - x + 3 = - 3 x + 6$

Case $2 :$ When $1 \leq x < 2$
$| x - 1 | + | x - 2 | + | x - 3 |$ becomes
$\Rightarrow f (x) = x - 1 - x + 2 - x + 3 = - x + 4$

Case $3 :$ When $2 \leq x < 3$
$| x - 1 | + | x - 2 | + | x - 3 |$ becomes
$\Rightarrow f (x) = x - 1 + x - 2 - x + 3 = x$

Case $4 :$ When $x \geq 3$
$| x - 1 | + | x - 2 | + | x - 3 |$ becomes
$\Rightarrow f (x) = x - 1 + x - 2 + x - 3 = 3 x - 6$

The graph is as shown for the above cases:
Thus, the minimum value of $f (x) = 2$

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