Let n ≥ 2 be a natural number and f : 0 ,   1 → ℝ be the function defined by f x = n 1 - 2 n x if 0 ≤ x ≤ 1 2 n 2 n 2 n x - 1 if 1 2 n ≤ x ≤ 3 4 n 4 n 1 - n x if 3 4 n ≤ x ≤ 1 n n n - 1 n x - 1 if 1 n ≤ x ≤ 1 If n is such that the area of the region bounded by the curves x = 0 ,   x = 1 ,   y = 0 and y = f x is 4 , then the maximum value of the function f is
A. A
Given, f x = n 1 - 2 n x , 0 ≤ x < 1 2 n 2 n 2 n x - 1 , 1 2 n ≤ x < 3 4 n 4 n 1 - n x , 3 4 n ≤ x ≤ 1 n n n - 1 n x - 1 , 1 n ≤ x ≤ 1 Here, x ∈ 0 ,   1 Now f x = n - 2 n 2 x is decreasing in 0 ,   1 2 n And f x = 4 n 2 x - 4 n increasing in 1 2 n ,   3 4 n And f x = 4 n - 4 n 2 x decreasing in 3 4 n ,   1 n And f x = n 2 x - n n - 1 increasing in 1 n ,   1 Now plotting the graph of above function we get, Also, f x ∈ 0 ,   n Now from above diagram area will be, A = 1 4 + 1 8 + 1 8 + n - 1 2 = 4 as   n ≥ 2 ⇒ 1 2 + n - 1 2 = 4 ⇒ n 2 = 4 ⇒ n = 8 So, f x m a x = n = 8
Related: Mathematics — Area Under Curves · All PYQ Banks