The minimum value of the expression $y = | x - 2 | + | x + 4 | + | x - 6 |$ is

6

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8

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10

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14

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Solution

The correct option is C $10$
$y = | x - 2 | + | x + 4 | + | x - 6 |$
Here the turning points are $- 4, 2, 6$

$| x - 2 | + | x + 4 | + | x - 6 | = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} - (x - 2) - (x + 4) - (x - 6), & x < - 4 - (x - 2) + (x + 4) - (x - 6), & - 4 \leq x < 2 (x - 2) + (x + 4) - (x - 6), & 2 \leq x < 6 (x - 2) + (x + 4) + (x - 6), & x \geq 6 \end{matrix}$

$\Rightarrow | x - 2 | + | x + 4 | + | x - 6 | = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} 4 - 3 x, & x < - 4 12 - x, & - 4 \leq x < 2 8 + x, & 2 \leq x < 6 3 x - 4, & x \geq 6 \end{matrix}$

Hence, the minimum value of the expression is $10$ .

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