Hi Vishwajeet.As per the equation given here, it is quite likely that f(x) = nx where n is some real number.This because, f(a^2 - b^2) = n(a^2 - b^2) = n(a - b)(a + b) = (a - b)(na + nb) = (a - b)(f(a) + f(b)).So, n = 1/101 based on the value of f(1).Now, the required k = sum(first 100 natural numbers) / (50 * 101) = 100 * 101 / (2 * 50 * 51) = 1

Yes , it will be though graphs. The best way to solve this is from options only.

Function image.png 16.96 KB

Function image.png 16.96 KB

Function Can we solve this without using option?Let 0 ≤ a ≤ x ≤ 100 f(x) =∣ x − a ∣ + ∣ x − 100 ∣ + ∣ x − a − 50 ∣. Then the maximum value of f(x) becomes 100 when a is equal to25100500

Function Can we solve this without using option?Let 0 ≤ a ≤ x ≤ 100 f(x) =∣ x − a ∣ + ∣ x − 100 ∣ + ∣ x − a − 50 ∣. Then the maximum value of f(x) becomes 100 when a is equal to25100500