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Let f : R → R be a differentiable function with f 0 = 1 and satisfying the equation f x + y = f x f ' y + f ' x f y for all x , y ∈ R . Then, the value of log e ⁡ f 4 is

Correct answer

0

Step-by-step solution

P x , y :   f x + y = f x f ' y + f ' x f y   ∀   x ,   y ∈ R P 0,0 :   f 0 = f 0 f ' 0 + f ' 0 f 0 ⇒ 1 = 2 f ' 0 ⇒ f ' 0 = 1 2 P x , 0 :   f x = f x . f ' 0 + f ' x . f 0 ⇒       f x = 1 2 f x + f ' x ⇒       f ' x = 1 2 f x ⇒       f x = e 1 2 x ⇒       ln ⁡ f 4 = 2

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