Given that f(x) = |x| + 2|x−1| + |x−2| + |x−4| + |x−6| + 2|x−10|, x ∈ (−∞, ∞) the minimum value of f(x) is _________.
Explanation:
Given, f(x) = |x| + 2|x−1| + |x−2| + |x−4| + |x−6| + 2|x−10|
f(x) can be written as sum of h(x) and g(x), where h(x) = |x| + |x−1| + |x−2| + |x−4| + |x−6| + |x−10| and g(x) = |x−1| + |x−10|
For h(x) critical points are 0, 1, 2, 4, 6 and 10. ∴ h(x) will be least when x is between 2 and 4.
For g(x) critical points are 1 and 10. ∴ g(x) will be least when x is between 1 and 10.
∴ Both h(x) and g(x) will be least when x is between 2 and 4.
Let us take x = 3.
Least value of h(x) = 3 + 2 + 1 + 1 + 3 + 7 = 17 Least value of g(x) = 2 + 7 = 9
⇒ Least value of f(x) = 17 + 9 = 26.
Hence, 26.
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