Let α be a non-zero real number. Suppose f : R → R is a differentiable function such that f 0 = 1 and lim x → − ∞ f x = 1 . If f ' x = α f x + 3 , for all x ∈ R , then f − log e 2 is equal to ________.

Step-by-step solution

Given: f 0 = 1 , lim x → − ∞ f x = 1 And f ' x − α · f x = 3 ⇒ d y d x − α · y = 3 Which is a linear differential equation, so ⇒ I . F . = e − α x Solution of the differential equation is given by, y e − α x = ∫ 3 · e − α x d x ⇒ f x ⋅ e − α x = 3 e − α x − α + c ⇒ f x = − 3 α + c · e α x Now, taking case I: when α > 0 and lim x → − ∞ f x = 1 we get, ⇒ 1 = − 3 α + c 0 α = − 3 (rejected) Case-II: α < 0 as lim x → − ∞ f x = 1 ⇒ c = 0 & − 3 α = 1 as e - - ∞ → ∞ , so c = 0 ⇒ α = − 3 ⇒ f x = 1 and also f 0 = 1 Hence, f