Let f : R → R and g : R → R be two functions defined by f x = log e x 2 + 1 - e - x + 1 and g x = 1 - 2 e 2 x e x · Then, for which of the following range of α , the inequality f g α - 1 2 3 > f g α - 5 3 holds?

Step-by-step solution

f x = log e x 2 + 1 - e - x + 1 ⇒ f ' x = 2 x x 2 + 1 + e - x > 0 , ∀ x ∈ R So f x increasing g x = e - x - 2 e x     ⇒   g ' x = - e - x + 2 e x < 0   ∀ x ∈ R So g x decreasing ⇒     f g x is decreasing ⇒     f g ( α - 1 ) 2 3 > f g α - 5 3 ⇒   α - 1 2 3 < α - 5 3 ⇒    α - 2 α - 3 < 0     ⇒ α ∈ 2 , 3