Discussion

Explanation:

Since a ≤ x ≤ 100
⇒ f(x) = x - a + -(x – 100) + |x - a - 50|
⇒ f(x) = x - a - x + 100 + |x - a - 50|
⇒ f(x) = 100 – a + |x - a - 50| ≤ 100

Option (a): a = 0
⇒ f(x) = 100 – 0 + |x - 0 - 50|
⇒ f(x) = 100 + |x - 50|
Here, when x = 0, f(x) > 100.
∴ This option is rejected.

Option (b): a = 25
⇒ f(x) = 100 – 25 + |x - 25 - 50|
⇒ f(x) = 75 + |x - 75|
Here, when x = 25, f(x) > 100.
∴ This option is rejected.

Option (c): a = 50
⇒ f(x) = 100 – 50 + |x - 50 - 50|
⇒ f(x) = 50 + |x - 100|
For any value of x ≥ 50, f(x) ≤ 100.
∴ This option is correct.

Option (d): a = 100
⇒ f(x) = 100 – 100 + |x - 100 - 50|
⇒ f(x) = |x - 150|
The only value x can take in this case is 100.
⇒ f(x) = 50
Here, the maximum value of f(x) is not 100.
∴ This option is rejected.

Hence, option (c).

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