If the function f : − ∞ , − 1 → a , b defined by f x = e x 3 − 3 x + 1 is one-one and onto, then the distance of the point P 2 b + 4 , a + 2 from the line x + e − 3 y = 4 is:

Step-by-step solution

Given: f x = e x 3 − 3 x + 1 ⇒ f ' x = e x 3 − 3 x + 1 × 3 x 2 - 3 ⇒ f ' x = 3 × e x 3 − 3 x + 1 × x - 1 × x + 1 So, for x ∈ ( - ∞ , - 1 ] f ' x > 0 , hence f x is increasing function in given domain, Now, finding a and b we get, ⇒ a = lim x → - ∞ e x 3 - 3 x + 1 ⇒ a = 0 And b = f - 1 ⇒ b = e - 1 + 3 + 1 ⇒ b = e 3 So, P 2 b + 4 , a + 2 ⇒ P ≡ 2 e 3 + 4 , 2 Now, finding the distance from the given line we get, ⇒ d = 2 e 3 + 4 + 2 e − 3 − 4 1 + e − 6 ⇒ d = 2 e 3 + e − 3 1 + e − 6 ⇒ d = 1 e - 3 2 1 + e − 6 1 + e − 6 ⇒