Discussion

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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