<p>The minimum value of the function \(y = |x+2| + |x-2| + |x-5| + |x-8| + |x-10|\) is:</p>
Step-by-Step Solution
Key Concept: The sum of absolute values |x-a₁| + |x-a₂| + ... + |x-aₙ| is minimized at the median of the points {a₁, a₂, ..., aₙ}. For odd number of points, the minimum occurs exactly at the median value.
<p><strong>Step 1:</strong> Rewrite the function in standard form: y = |x-(-2)| + |x-2| + |x-5| + |x-8| + |x-10|</p><p><strong>Step 2:</strong> Identify the points: -2, 2, 5, 8, 10 (5 points total)</p><p><strong>Step 3:</strong> Arrange in order: -2, 2, 5, 8, 10. The median is the middle (3rd) value = 5</p><p><strong>Step 4:</strong> The sum of absolute deviations is minimized when x = median = 5</p><p><strong>Step 5:</strong> Calculate minimum value at x = 5:</p><p>y = |5+2| + |5-2| + |5-5| + |5-8| + |5-10|</p><p>y = |7| + |3| + |0| + |-3| + |-5|</p><p>y = 7 + 3 + 0 + 3 + 5 = 18</p><p>∴ Answer: C (Minimum value = 18)</p>
Correct Answer: C