If f : [ 0 , 2 ) → ℝ is defined by f x = 1 + 2 x k for 0 ≤ x < 1 k x for 1 ≤ x < 2 , where k > 0 and f is such that lim x → 1 - f x = lim x → 1 + f x , then the value of k 2 is

Step-by-step solution

Given: f x = 1 + 2 x k  for  0 ≤ x < 1 k x  for  1 ≤ x < 2 Now, lim x → 1 - f x = lim x → 1 + f x ⇒ 1 + 2 k = k ⇒ k + 2 = k 2 ⇒ k 2 - k - 2 = 0 ⇒ k 2 - 2 k + k - 2 = 0 ⇒ k + 1 k - 2 = 0 ⇒ k = 2   and   - 1 But, k > 0 Hence, k = 2 ⇒ k 2 = 4 .