The minimum value of f(x) = |3 - x| + |2 + x| + |5 - x| is equal to __________.
Explanation:
Given, |3 - x| + |2 + x| + |5 - x|
= |x - 3| + |x + 2| + |x - 5|
= |x + 2| + |x - 3| + |x - 5|
Here, the critical points are -2, 3 and 5.
The minimum possible value of |x + 2| + |x - 5| is when x is between -2 and 5. The minimum possible value of |x - 3| is when x = 3.
∴ x = 3 minimises |x + 2| + |x - 3| + |x - 5|.
⇒ Minimum value of |x + 2| + |x - 3| + |x - 5| = |3 + 2| + |3 - 3| + |3 - 5|
= 5 + 0 + 2 = 7
Hence, 7.
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