Question: The function f(x) = |x − 2| + |2.5 − x| + |3.6 − x|, where x is a real number, attains a minimum at
Explanation:
f(x) = |x − 2| + |2.5 − x| + |3.6 − x| = g(x) + |2.5 – x|, where g(x) = |x − 2| + |3.6 − x|
When 2 ≤ x ≤ 3.6, g(x) attains a fixed value.
This happens as in this range |x – 2| = x – 2 and |3.6 – x| = 3.6 – x
∴ |x − 2| + |3.6 − x| = x – 2 + 3.6 – x = 1.6
Also, as –x > –2, 3.6 – x > 3.6 – 2
∴ 3.6 – x > 1.6
∴ |x − 2| + |3.6 − x| > 1.6
Similarly, when x > 3.6,
|3.6 – x| > 0 and |x – 2| > 1.6
∴ |x − 2| + |3.6 − x| > 1.6
As f(x) = g(x) + |2.5 – x|,
∴ f(x) attains the minimum value when 2 ≤ x ≤ 3.6 and |2.5 – x| is minimum.
This happens when x = 2.5
∴ f(x) attains minimum when x = 2.5
Hence, option (b).Alternatively,
f(x) = |x − 2| + |2.5 − x| + |3.6 − x|
Substituting the value of x from the given options in the function,
when x = 2.3
f(x) = 0.3 + 0.2 + 1.3 = 1.8
when x = 2.5
f(x) = 0.5 + 0 + 1.1 = 1.6
when x = 2.7
f(x) = 0.7 + 0.2 + 0.9 = 1.8
Substituting any arbitrary real values of x in f(x),
when x = 2
f(x) = 0 + 0.5 + 1.6 = 2.1
when x = 3
f(x) = 1 + 0.5 + 0.6 = 2.1
For any other value of x, f(x) will be greater than 1.6
Hence, f(x) is minimum when x = 2.5
Hence, option (b).