Let f : - 1 , 1 → R be a differentiable function satisfying f ' x 4 = 16 f x 2 for all x ∈ - 1 , 1 f 0 = 0 The number of such functions is

Step-by-step solution

It is given that, f ' x 4 = 16 f x 2 , ∀ x ∈ - 1 , 1 ⇒ f ' x 2 = ± 4 f x Case I If f x ≥ 0 ∀ x ∈ - 1 , 1 Then, f ' x = ± 2 f x ⇒ f ' x f x = ± 2 ⇒ 2 f x = ± 2 x ,    ∵ f 0 = 0 ⇒ f x = x 2 ∀ x ∈ - 1 , 1 Case II If f x < 0 ∀ x ∈ - 1 , 1 Then, f ' x = ± 2 - f x ∴   f x = - x 2 ∀ x ∈ - 1 , 1 ∴   f 1 x = x 2 , 0 ≤ x < 1 x 2 , - 1 <